3.17.64 \(\int \frac {(d+e x)^{15/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1664]

Optimal. Leaf size=253 \[ \frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}} \]

[Out]

3003/128*e^5*(-a*e+b*d)*(e*x+d)^(3/2)/b^7+9009/640*e^5*(e*x+d)^(5/2)/b^6-1287/128*e^4*(e*x+d)^(7/2)/b^5/(b*x+a
)-143/64*e^3*(e*x+d)^(9/2)/b^4/(b*x+a)^2-13/16*e^2*(e*x+d)^(11/2)/b^3/(b*x+a)^3-3/8*e*(e*x+d)^(13/2)/b^2/(b*x+
a)^4-1/5*(e*x+d)^(15/2)/b/(b*x+a)^5-9009/128*e^5*(-a*e+b*d)^(5/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/
2))/b^(17/2)+9009/128*e^5*(-a*e+b*d)^2*(e*x+d)^(1/2)/b^8

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Rubi [A]
time = 0.12, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \begin {gather*} -\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}+\frac {9009 e^5 \sqrt {d+e x} (b d-a e)^2}{128 b^8}+\frac {3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(9009*e^5*(b*d - a*e)^2*Sqrt[d + e*x])/(128*b^8) + (3003*e^5*(b*d - a*e)*(d + e*x)^(3/2))/(128*b^7) + (9009*e^
5*(d + e*x)^(5/2))/(640*b^6) - (1287*e^4*(d + e*x)^(7/2))/(128*b^5*(a + b*x)) - (143*e^3*(d + e*x)^(9/2))/(64*
b^4*(a + b*x)^2) - (13*e^2*(d + e*x)^(11/2))/(16*b^3*(a + b*x)^3) - (3*e*(d + e*x)^(13/2))/(8*b^2*(a + b*x)^4)
- (d + e*x)^(15/2)/(5*b*(a + b*x)^5) - (9009*e^5*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(128*b^(17/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{15/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {(3 e) \int \frac {(d+e x)^{13/2}}{(a+b x)^5} \, dx}{2 b}\\ &=-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (39 e^2\right ) \int \frac {(d+e x)^{11/2}}{(a+b x)^4} \, dx}{16 b^2}\\ &=-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (143 e^3\right ) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{32 b^3}\\ &=-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (1287 e^4\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{256 b^5}\\ &=\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^6}\\ &=\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^7}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^8}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^4 (b d-a e)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^8}\\ &=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}\\ \end {align*}

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Mathematica [A]
time = 2.05, size = 434, normalized size = 1.72 \begin {gather*} -\frac {\sqrt {d+e x} \left (-45045 a^7 e^7+105105 a^6 b e^6 (d-2 e x)-3003 a^5 b^2 e^5 \left (23 d^2-164 d e x+128 e^2 x^2\right )+2145 a^4 b^3 e^4 \left (3 d^3-152 d^2 e x+422 d e^2 x^2-158 e^3 x^3\right )+715 a^3 b^4 e^3 \left (2 d^4+44 d^3 e x-846 d^2 e^2 x^2+1124 d e^3 x^3-193 e^4 x^4\right )+65 a^2 b^5 e^2 \left (8 d^5+106 d^4 e x+938 d^3 e^2 x^2-8368 d^2 e^3 x^3+5089 d e^4 x^4-256 e^5 x^5\right )+5 a b^6 e \left (48 d^6+496 d^5 e x+2618 d^4 e^2 x^2+11620 d^3 e^3 x^3-45677 d^2 e^4 x^4+8192 d e^5 x^5+256 e^6 x^6\right )+b^7 \left (128 d^7+1136 d^6 e x+4648 d^5 e^2 x^2+12110 d^4 e^3 x^3+26635 d^3 e^4 x^4-29696 d^2 e^5 x^5-3072 d e^6 x^6-256 e^7 x^7\right )\right )}{640 b^8 (a+b x)^5}-\frac {9009 e^5 (-b d+a e)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{17/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(15/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/640*(Sqrt[d + e*x]*(-45045*a^7*e^7 + 105105*a^6*b*e^6*(d - 2*e*x) - 3003*a^5*b^2*e^5*(23*d^2 - 164*d*e*x +
128*e^2*x^2) + 2145*a^4*b^3*e^4*(3*d^3 - 152*d^2*e*x + 422*d*e^2*x^2 - 158*e^3*x^3) + 715*a^3*b^4*e^3*(2*d^4 +
44*d^3*e*x - 846*d^2*e^2*x^2 + 1124*d*e^3*x^3 - 193*e^4*x^4) + 65*a^2*b^5*e^2*(8*d^5 + 106*d^4*e*x + 938*d^3*
e^2*x^2 - 8368*d^2*e^3*x^3 + 5089*d*e^4*x^4 - 256*e^5*x^5) + 5*a*b^6*e*(48*d^6 + 496*d^5*e*x + 2618*d^4*e^2*x^
2 + 11620*d^3*e^3*x^3 - 45677*d^2*e^4*x^4 + 8192*d*e^5*x^5 + 256*e^6*x^6) + b^7*(128*d^7 + 1136*d^6*e*x + 4648
*d^5*e^2*x^2 + 12110*d^4*e^3*x^3 + 26635*d^3*e^4*x^4 - 29696*d^2*e^5*x^5 - 3072*d*e^6*x^6 - 256*e^7*x^7)))/(b^
8*(a + b*x)^5) - (9009*e^5*(-(b*d) + a*e)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(17
/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(568\) vs. \(2(209)=418\).
time = 0.74, size = 569, normalized size = 2.25

method	result	size
derivativedivides	\(2 e^{5} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-2 a b e \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}+21 a^{2} e^{2} \sqrt {e x +d}-42 a b d e \sqrt {e x +d}+21 b^{2} d^{2} \sqrt {e x +d}}{b^{8}}-\frac {\frac {\left (-\frac {5327}{256} a^{3} e^{3} b^{4}+\frac {15981}{256} a^{2} b^{5} d \,e^{2}-\frac {15981}{256} a \,b^{6} d^{2} e +\frac {5327}{256} b^{7} d^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {9443 b^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {1001}{10} a^{5} e^{5} b^{2}+\frac {1001}{2} a^{4} b^{3} d \,e^{4}-1001 a^{3} b^{4} d^{2} e^{3}+1001 a^{2} b^{5} d^{3} e^{2}-\frac {1001}{2} a \,b^{6} d^{4} e +\frac {1001}{10} b^{7} d^{5}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {7837}{128} a^{6} e^{6} b +\frac {23511}{64} a^{5} b^{2} d \,e^{5}-\frac {117555}{128} a^{4} b^{3} d^{2} e^{4}+\frac {39185}{32} a^{3} b^{4} d^{3} e^{3}-\frac {117555}{128} a^{2} b^{5} d^{4} e^{2}+\frac {23511}{64} a \,b^{6} d^{5} e -\frac {7837}{128} b^{7} d^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3633}{256} a^{7} e^{7}+\frac {25431}{256} a^{6} b d \,e^{6}-\frac {76293}{256} a^{5} b^{2} d^{2} e^{5}+\frac {127155}{256} a^{4} b^{3} d^{3} e^{4}-\frac {127155}{256} a^{3} b^{4} d^{4} e^{3}+\frac {76293}{256} a^{2} b^{5} d^{5} e^{2}-\frac {25431}{256} a \,b^{6} d^{6} e +\frac {3633}{256} b^{7} d^{7}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {9009 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}}{b^{8}}\right )\)	\(569\)
default	\(2 e^{5} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-2 a b e \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}+21 a^{2} e^{2} \sqrt {e x +d}-42 a b d e \sqrt {e x +d}+21 b^{2} d^{2} \sqrt {e x +d}}{b^{8}}-\frac {\frac {\left (-\frac {5327}{256} a^{3} e^{3} b^{4}+\frac {15981}{256} a^{2} b^{5} d \,e^{2}-\frac {15981}{256} a \,b^{6} d^{2} e +\frac {5327}{256} b^{7} d^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {9443 b^{3} \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {1001}{10} a^{5} e^{5} b^{2}+\frac {1001}{2} a^{4} b^{3} d \,e^{4}-1001 a^{3} b^{4} d^{2} e^{3}+1001 a^{2} b^{5} d^{3} e^{2}-\frac {1001}{2} a \,b^{6} d^{4} e +\frac {1001}{10} b^{7} d^{5}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {7837}{128} a^{6} e^{6} b +\frac {23511}{64} a^{5} b^{2} d \,e^{5}-\frac {117555}{128} a^{4} b^{3} d^{2} e^{4}+\frac {39185}{32} a^{3} b^{4} d^{3} e^{3}-\frac {117555}{128} a^{2} b^{5} d^{4} e^{2}+\frac {23511}{64} a \,b^{6} d^{5} e -\frac {7837}{128} b^{7} d^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3633}{256} a^{7} e^{7}+\frac {25431}{256} a^{6} b d \,e^{6}-\frac {76293}{256} a^{5} b^{2} d^{2} e^{5}+\frac {127155}{256} a^{4} b^{3} d^{3} e^{4}-\frac {127155}{256} a^{3} b^{4} d^{4} e^{3}+\frac {76293}{256} a^{2} b^{5} d^{5} e^{2}-\frac {25431}{256} a \,b^{6} d^{6} e +\frac {3633}{256} b^{7} d^{7}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{5}}+\frac {9009 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{256 \sqrt {b \left (a e -b d \right )}}}{b^{8}}\right )\)	\(569\)
risch	\(\text {Expression too large to display}\)	\(1128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(1/b^8*(1/5*b^2*(e*x+d)^(5/2)-2*a*b*e*(e*x+d)^(3/2)+2*b^2*d*(e*x+d)^(3/2)+21*a^2*e^2*(e*x+d)^(1/2)-42*a*
b*d*e*(e*x+d)^(1/2)+21*b^2*d^2*(e*x+d)^(1/2))-1/b^8*(((-5327/256*a^3*e^3*b^4+15981/256*a^2*b^5*d*e^2-15981/256
*a*b^6*d^2*e+5327/256*b^7*d^3)*(e*x+d)^(9/2)-9443/128*b^3*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3
*e+b^4*d^4)*(e*x+d)^(7/2)+(-1001/10*a^5*e^5*b^2+1001/2*a^4*b^3*d*e^4-1001*a^3*b^4*d^2*e^3+1001*a^2*b^5*d^3*e^2
-1001/2*a*b^6*d^4*e+1001/10*b^7*d^5)*(e*x+d)^(5/2)+(-7837/128*a^6*e^6*b+23511/64*a^5*b^2*d*e^5-117555/128*a^4*
b^3*d^2*e^4+39185/32*a^3*b^4*d^3*e^3-117555/128*a^2*b^5*d^4*e^2+23511/64*a*b^6*d^5*e-7837/128*b^7*d^6)*(e*x+d)
^(3/2)+(-3633/256*a^7*e^7+25431/256*a^6*b*d*e^6-76293/256*a^5*b^2*d^2*e^5+127155/256*a^4*b^3*d^3*e^4-127155/25
6*a^3*b^4*d^4*e^3+76293/256*a^2*b^5*d^5*e^2-25431/256*a*b^6*d^6*e+3633/256*b^7*d^7)*(e*x+d)^(1/2))/((e*x+d)*b+
a*e-b*d)^5+9009/256*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(
b*(a*e-b*d))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (216) = 432\).
time = 3.93, size = 1502, normalized size = 5.94 \begin {gather*} \left [\frac {45045 \, {\left ({\left (a^{2} b^{5} x^{5} + 5 \, a^{3} b^{4} x^{4} + 10 \, a^{4} b^{3} x^{3} + 10 \, a^{5} b^{2} x^{2} + 5 \, a^{6} b x + a^{7}\right )} e^{7} - 2 \, {\left (a b^{6} d x^{5} + 5 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 10 \, a^{4} b^{3} d x^{2} + 5 \, a^{5} b^{2} d x + a^{6} b d\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (128 \, b^{7} d^{7} - {\left (256 \, b^{7} x^{7} - 1280 \, a b^{6} x^{6} + 16640 \, a^{2} b^{5} x^{5} + 137995 \, a^{3} b^{4} x^{4} + 338910 \, a^{4} b^{3} x^{3} + 384384 \, a^{5} b^{2} x^{2} + 210210 \, a^{6} b x + 45045 \, a^{7}\right )} e^{7} - {\left (3072 \, b^{7} d x^{6} - 40960 \, a b^{6} d x^{5} - 330785 \, a^{2} b^{5} d x^{4} - 803660 \, a^{3} b^{4} d x^{3} - 905190 \, a^{4} b^{3} d x^{2} - 492492 \, a^{5} b^{2} d x - 105105 \, a^{6} b d\right )} e^{6} - {\left (29696 \, b^{7} d^{2} x^{5} + 228385 \, a b^{6} d^{2} x^{4} + 543920 \, a^{2} b^{5} d^{2} x^{3} + 604890 \, a^{3} b^{4} d^{2} x^{2} + 326040 \, a^{4} b^{3} d^{2} x + 69069 \, a^{5} b^{2} d^{2}\right )} e^{5} + 5 \, {\left (5327 \, b^{7} d^{3} x^{4} + 11620 \, a b^{6} d^{3} x^{3} + 12194 \, a^{2} b^{5} d^{3} x^{2} + 6292 \, a^{3} b^{4} d^{3} x + 1287 \, a^{4} b^{3} d^{3}\right )} e^{4} + 10 \, {\left (1211 \, b^{7} d^{4} x^{3} + 1309 \, a b^{6} d^{4} x^{2} + 689 \, a^{2} b^{5} d^{4} x + 143 \, a^{3} b^{4} d^{4}\right )} e^{3} + 8 \, {\left (581 \, b^{7} d^{5} x^{2} + 310 \, a b^{6} d^{5} x + 65 \, a^{2} b^{5} d^{5}\right )} e^{2} + 16 \, {\left (71 \, b^{7} d^{6} x + 15 \, a b^{6} d^{6}\right )} e\right )} \sqrt {x e + d}}{1280 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}}, -\frac {45045 \, {\left ({\left (a^{2} b^{5} x^{5} + 5 \, a^{3} b^{4} x^{4} + 10 \, a^{4} b^{3} x^{3} + 10 \, a^{5} b^{2} x^{2} + 5 \, a^{6} b x + a^{7}\right )} e^{7} - 2 \, {\left (a b^{6} d x^{5} + 5 \, a^{2} b^{5} d x^{4} + 10 \, a^{3} b^{4} d x^{3} + 10 \, a^{4} b^{3} d x^{2} + 5 \, a^{5} b^{2} d x + a^{6} b d\right )} e^{6} + {\left (b^{7} d^{2} x^{5} + 5 \, a b^{6} d^{2} x^{4} + 10 \, a^{2} b^{5} d^{2} x^{3} + 10 \, a^{3} b^{4} d^{2} x^{2} + 5 \, a^{4} b^{3} d^{2} x + a^{5} b^{2} d^{2}\right )} e^{5}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (128 \, b^{7} d^{7} - {\left (256 \, b^{7} x^{7} - 1280 \, a b^{6} x^{6} + 16640 \, a^{2} b^{5} x^{5} + 137995 \, a^{3} b^{4} x^{4} + 338910 \, a^{4} b^{3} x^{3} + 384384 \, a^{5} b^{2} x^{2} + 210210 \, a^{6} b x + 45045 \, a^{7}\right )} e^{7} - {\left (3072 \, b^{7} d x^{6} - 40960 \, a b^{6} d x^{5} - 330785 \, a^{2} b^{5} d x^{4} - 803660 \, a^{3} b^{4} d x^{3} - 905190 \, a^{4} b^{3} d x^{2} - 492492 \, a^{5} b^{2} d x - 105105 \, a^{6} b d\right )} e^{6} - {\left (29696 \, b^{7} d^{2} x^{5} + 228385 \, a b^{6} d^{2} x^{4} + 543920 \, a^{2} b^{5} d^{2} x^{3} + 604890 \, a^{3} b^{4} d^{2} x^{2} + 326040 \, a^{4} b^{3} d^{2} x + 69069 \, a^{5} b^{2} d^{2}\right )} e^{5} + 5 \, {\left (5327 \, b^{7} d^{3} x^{4} + 11620 \, a b^{6} d^{3} x^{3} + 12194 \, a^{2} b^{5} d^{3} x^{2} + 6292 \, a^{3} b^{4} d^{3} x + 1287 \, a^{4} b^{3} d^{3}\right )} e^{4} + 10 \, {\left (1211 \, b^{7} d^{4} x^{3} + 1309 \, a b^{6} d^{4} x^{2} + 689 \, a^{2} b^{5} d^{4} x + 143 \, a^{3} b^{4} d^{4}\right )} e^{3} + 8 \, {\left (581 \, b^{7} d^{5} x^{2} + 310 \, a b^{6} d^{5} x + 65 \, a^{2} b^{5} d^{5}\right )} e^{2} + 16 \, {\left (71 \, b^{7} d^{6} x + 15 \, a b^{6} d^{6}\right )} e\right )} \sqrt {x e + d}}{640 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(45045*((a^2*b^5*x^5 + 5*a^3*b^4*x^4 + 10*a^4*b^3*x^3 + 10*a^5*b^2*x^2 + 5*a^6*b*x + a^7)*e^7 - 2*(a*b
^6*d*x^5 + 5*a^2*b^5*d*x^4 + 10*a^3*b^4*d*x^3 + 10*a^4*b^3*d*x^2 + 5*a^5*b^2*d*x + a^6*b*d)*e^6 + (b^7*d^2*x^5
+ 5*a*b^6*d^2*x^4 + 10*a^2*b^5*d^2*x^3 + 10*a^3*b^4*d^2*x^2 + 5*a^4*b^3*d^2*x + a^5*b^2*d^2)*e^5)*sqrt((b*d -
a*e)/b)*log((2*b*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(128*b^7*d^7 - (256*
b^7*x^7 - 1280*a*b^6*x^6 + 16640*a^2*b^5*x^5 + 137995*a^3*b^4*x^4 + 338910*a^4*b^3*x^3 + 384384*a^5*b^2*x^2 +
210210*a^6*b*x + 45045*a^7)*e^7 - (3072*b^7*d*x^6 - 40960*a*b^6*d*x^5 - 330785*a^2*b^5*d*x^4 - 803660*a^3*b^4*
d*x^3 - 905190*a^4*b^3*d*x^2 - 492492*a^5*b^2*d*x - 105105*a^6*b*d)*e^6 - (29696*b^7*d^2*x^5 + 228385*a*b^6*d^
2*x^4 + 543920*a^2*b^5*d^2*x^3 + 604890*a^3*b^4*d^2*x^2 + 326040*a^4*b^3*d^2*x + 69069*a^5*b^2*d^2)*e^5 + 5*(5
327*b^7*d^3*x^4 + 11620*a*b^6*d^3*x^3 + 12194*a^2*b^5*d^3*x^2 + 6292*a^3*b^4*d^3*x + 1287*a^4*b^3*d^3)*e^4 + 1
0*(1211*b^7*d^4*x^3 + 1309*a*b^6*d^4*x^2 + 689*a^2*b^5*d^4*x + 143*a^3*b^4*d^4)*e^3 + 8*(581*b^7*d^5*x^2 + 310
*a*b^6*d^5*x + 65*a^2*b^5*d^5)*e^2 + 16*(71*b^7*d^6*x + 15*a*b^6*d^6)*e)*sqrt(x*e + d))/(b^13*x^5 + 5*a*b^12*x
^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8), -1/640*(45045*((a^2*b^5*x^5 + 5*a^3*b^4*x^4 +
10*a^4*b^3*x^3 + 10*a^5*b^2*x^2 + 5*a^6*b*x + a^7)*e^7 - 2*(a*b^6*d*x^5 + 5*a^2*b^5*d*x^4 + 10*a^3*b^4*d*x^3
+ 10*a^4*b^3*d*x^2 + 5*a^5*b^2*d*x + a^6*b*d)*e^6 + (b^7*d^2*x^5 + 5*a*b^6*d^2*x^4 + 10*a^2*b^5*d^2*x^3 + 10*a
^3*b^4*d^2*x^2 + 5*a^4*b^3*d^2*x + a^5*b^2*d^2)*e^5)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(x*e + d)*b*sqrt(-(b*d -
a*e)/b)/(b*d - a*e)) + (128*b^7*d^7 - (256*b^7*x^7 - 1280*a*b^6*x^6 + 16640*a^2*b^5*x^5 + 137995*a^3*b^4*x^4
+ 338910*a^4*b^3*x^3 + 384384*a^5*b^2*x^2 + 210210*a^6*b*x + 45045*a^7)*e^7 - (3072*b^7*d*x^6 - 40960*a*b^6*d*
x^5 - 330785*a^2*b^5*d*x^4 - 803660*a^3*b^4*d*x^3 - 905190*a^4*b^3*d*x^2 - 492492*a^5*b^2*d*x - 105105*a^6*b*d
)*e^6 - (29696*b^7*d^2*x^5 + 228385*a*b^6*d^2*x^4 + 543920*a^2*b^5*d^2*x^3 + 604890*a^3*b^4*d^2*x^2 + 326040*a
^4*b^3*d^2*x + 69069*a^5*b^2*d^2)*e^5 + 5*(5327*b^7*d^3*x^4 + 11620*a*b^6*d^3*x^3 + 12194*a^2*b^5*d^3*x^2 + 62
92*a^3*b^4*d^3*x + 1287*a^4*b^3*d^3)*e^4 + 10*(1211*b^7*d^4*x^3 + 1309*a*b^6*d^4*x^2 + 689*a^2*b^5*d^4*x + 143
*a^3*b^4*d^4)*e^3 + 8*(581*b^7*d^5*x^2 + 310*a*b^6*d^5*x + 65*a^2*b^5*d^5)*e^2 + 16*(71*b^7*d^6*x + 15*a*b^6*d
^6)*e)*sqrt(x*e + d))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(15/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (216) = 432\).
time = 1.48, size = 785, normalized size = 3.10 \begin {gather*} \frac {9009 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{8}} - \frac {26635 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d^{3} e^{5} - 94430 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{4} e^{5} + 128128 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{5} e^{5} - 78370 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{6} e^{5} + 18165 \, \sqrt {x e + d} b^{7} d^{7} e^{5} - 79905 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} d^{2} e^{6} + 377720 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d^{3} e^{6} - 640640 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{4} e^{6} + 470220 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{5} e^{6} - 127155 \, \sqrt {x e + d} a b^{6} d^{6} e^{6} + 79905 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{2} b^{5} d e^{7} - 566580 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} d^{2} e^{7} + 1281280 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{3} e^{7} - 1175550 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{4} e^{7} + 381465 \, \sqrt {x e + d} a^{2} b^{5} d^{5} e^{7} - 26635 \, {\left (x e + d\right )}^{\frac {9}{2}} a^{3} b^{4} e^{8} + 377720 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{3} b^{4} d e^{8} - 1281280 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} d^{2} e^{8} + 1567400 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{3} e^{8} - 635775 \, \sqrt {x e + d} a^{3} b^{4} d^{4} e^{8} - 94430 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{4} b^{3} e^{9} + 640640 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{4} b^{3} d e^{9} - 1175550 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} d^{2} e^{9} + 635775 \, \sqrt {x e + d} a^{4} b^{3} d^{3} e^{9} - 128128 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{5} b^{2} e^{10} + 470220 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{5} b^{2} d e^{10} - 381465 \, \sqrt {x e + d} a^{5} b^{2} d^{2} e^{10} - 78370 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{6} b e^{11} + 127155 \, \sqrt {x e + d} a^{6} b d e^{11} - 18165 \, \sqrt {x e + d} a^{7} e^{12}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{8}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} b^{24} e^{5} + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{24} d e^{5} + 105 \, \sqrt {x e + d} b^{24} d^{2} e^{5} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{23} e^{6} - 210 \, \sqrt {x e + d} a b^{23} d e^{6} + 105 \, \sqrt {x e + d} a^{2} b^{22} e^{7}\right )}}{5 \, b^{30}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(15/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

9009/128*(b^3*d^3*e^5 - 3*a*b^2*d^2*e^6 + 3*a^2*b*d*e^7 - a^3*e^8)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e)
)/(sqrt(-b^2*d + a*b*e)*b^8) - 1/640*(26635*(x*e + d)^(9/2)*b^7*d^3*e^5 - 94430*(x*e + d)^(7/2)*b^7*d^4*e^5 +
128128*(x*e + d)^(5/2)*b^7*d^5*e^5 - 78370*(x*e + d)^(3/2)*b^7*d^6*e^5 + 18165*sqrt(x*e + d)*b^7*d^7*e^5 - 799
05*(x*e + d)^(9/2)*a*b^6*d^2*e^6 + 377720*(x*e + d)^(7/2)*a*b^6*d^3*e^6 - 640640*(x*e + d)^(5/2)*a*b^6*d^4*e^6
+ 470220*(x*e + d)^(3/2)*a*b^6*d^5*e^6 - 127155*sqrt(x*e + d)*a*b^6*d^6*e^6 + 79905*(x*e + d)^(9/2)*a^2*b^5*d
*e^7 - 566580*(x*e + d)^(7/2)*a^2*b^5*d^2*e^7 + 1281280*(x*e + d)^(5/2)*a^2*b^5*d^3*e^7 - 1175550*(x*e + d)^(3
/2)*a^2*b^5*d^4*e^7 + 381465*sqrt(x*e + d)*a^2*b^5*d^5*e^7 - 26635*(x*e + d)^(9/2)*a^3*b^4*e^8 + 377720*(x*e +
d)^(7/2)*a^3*b^4*d*e^8 - 1281280*(x*e + d)^(5/2)*a^3*b^4*d^2*e^8 + 1567400*(x*e + d)^(3/2)*a^3*b^4*d^3*e^8 -
635775*sqrt(x*e + d)*a^3*b^4*d^4*e^8 - 94430*(x*e + d)^(7/2)*a^4*b^3*e^9 + 640640*(x*e + d)^(5/2)*a^4*b^3*d*e^
9 - 1175550*(x*e + d)^(3/2)*a^4*b^3*d^2*e^9 + 635775*sqrt(x*e + d)*a^4*b^3*d^3*e^9 - 128128*(x*e + d)^(5/2)*a^
5*b^2*e^10 + 470220*(x*e + d)^(3/2)*a^5*b^2*d*e^10 - 381465*sqrt(x*e + d)*a^5*b^2*d^2*e^10 - 78370*(x*e + d)^(
3/2)*a^6*b*e^11 + 127155*sqrt(x*e + d)*a^6*b*d*e^11 - 18165*sqrt(x*e + d)*a^7*e^12)/(((x*e + d)*b - b*d + a*e)
^5*b^8) + 2/5*((x*e + d)^(5/2)*b^24*e^5 + 10*(x*e + d)^(3/2)*b^24*d*e^5 + 105*sqrt(x*e + d)*b^24*d^2*e^5 - 10*
(x*e + d)^(3/2)*a*b^23*e^6 - 210*sqrt(x*e + d)*a*b^23*d*e^6 + 105*sqrt(x*e + d)*a^2*b^22*e^7)/b^30

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Mupad [B]
time = 0.83, size = 846, normalized size = 3.34 \begin {gather*} \left (\frac {2\,e^5\,{\left (6\,b^6\,d-6\,a\,b^5\,e\right )}^2}{b^{18}}-\frac {30\,e^5\,{\left (a\,e-b\,d\right )}^2}{b^8}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {3633\,a^7\,e^{12}}{128}-\frac {25431\,a^6\,b\,d\,e^{11}}{128}+\frac {76293\,a^5\,b^2\,d^2\,e^{10}}{128}-\frac {127155\,a^4\,b^3\,d^3\,e^9}{128}+\frac {127155\,a^3\,b^4\,d^4\,e^8}{128}-\frac {76293\,a^2\,b^5\,d^5\,e^7}{128}+\frac {25431\,a\,b^6\,d^6\,e^6}{128}-\frac {3633\,b^7\,d^7\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1001\,a^5\,b^2\,e^{10}}{5}-1001\,a^4\,b^3\,d\,e^9+2002\,a^3\,b^4\,d^2\,e^8-2002\,a^2\,b^5\,d^3\,e^7+1001\,a\,b^6\,d^4\,e^6-\frac {1001\,b^7\,d^5\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {7837\,a^6\,b\,e^{11}}{64}-\frac {23511\,a^5\,b^2\,d\,e^{10}}{32}+\frac {117555\,a^4\,b^3\,d^2\,e^9}{64}-\frac {39185\,a^3\,b^4\,d^3\,e^8}{16}+\frac {117555\,a^2\,b^5\,d^4\,e^7}{64}-\frac {23511\,a\,b^6\,d^5\,e^6}{32}+\frac {7837\,b^7\,d^6\,e^5}{64}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {5327\,a^3\,b^4\,e^8}{128}-\frac {15981\,a^2\,b^5\,d\,e^7}{128}+\frac {15981\,a\,b^6\,d^2\,e^6}{128}-\frac {5327\,b^7\,d^3\,e^5}{128}\right )+{\left (d+e\,x\right )}^{7/2}\,\left (\frac {9443\,a^4\,b^3\,e^9}{64}-\frac {9443\,a^3\,b^4\,d\,e^8}{16}+\frac {28329\,a^2\,b^5\,d^2\,e^7}{32}-\frac {9443\,a\,b^6\,d^3\,e^6}{16}+\frac {9443\,b^7\,d^4\,e^5}{64}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^9\,e^4-20\,a^3\,b^{10}\,d\,e^3+30\,a^2\,b^{11}\,d^2\,e^2-20\,a\,b^{12}\,d^3\,e+5\,b^{13}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^{10}\,e^3+30\,a^2\,b^{11}\,d\,e^2-30\,a\,b^{12}\,d^2\,e+10\,b^{13}\,d^3\right )+b^{13}\,{\left (d+e\,x\right )}^5-\left (5\,b^{13}\,d-5\,a\,b^{12}\,e\right )\,{\left (d+e\,x\right )}^4-b^{13}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{11}\,e^2-20\,a\,b^{12}\,d\,e+10\,b^{13}\,d^2\right )+a^5\,b^8\,e^5-5\,a^4\,b^9\,d\,e^4-10\,a^2\,b^{11}\,d^3\,e^2+10\,a^3\,b^{10}\,d^2\,e^3+5\,a\,b^{12}\,d^4\,e}+\frac {2\,e^5\,{\left (d+e\,x\right )}^{5/2}}{5\,b^6}+\frac {2\,e^5\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^{12}}-\frac {9009\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^8-3\,a^2\,b\,d\,e^7+3\,a\,b^2\,d^2\,e^6-b^3\,d^3\,e^5}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{128\,b^{17/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(15/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((2*e^5*(6*b^6*d - 6*a*b^5*e)^2)/b^18 - (30*e^5*(a*e - b*d)^2)/b^8)*(d + e*x)^(1/2) + ((d + e*x)^(1/2)*((3633*
a^7*e^12)/128 - (3633*b^7*d^7*e^5)/128 + (25431*a*b^6*d^6*e^6)/128 - (76293*a^2*b^5*d^5*e^7)/128 + (127155*a^3
*b^4*d^4*e^8)/128 - (127155*a^4*b^3*d^3*e^9)/128 + (76293*a^5*b^2*d^2*e^10)/128 - (25431*a^6*b*d*e^11)/128) +
(d + e*x)^(5/2)*((1001*a^5*b^2*e^10)/5 - (1001*b^7*d^5*e^5)/5 + 1001*a*b^6*d^4*e^6 - 1001*a^4*b^3*d*e^9 - 2002
*a^2*b^5*d^3*e^7 + 2002*a^3*b^4*d^2*e^8) + (d + e*x)^(3/2)*((7837*a^6*b*e^11)/64 + (7837*b^7*d^6*e^5)/64 - (23
511*a*b^6*d^5*e^6)/32 - (23511*a^5*b^2*d*e^10)/32 + (117555*a^2*b^5*d^4*e^7)/64 - (39185*a^3*b^4*d^3*e^8)/16 +
(117555*a^4*b^3*d^2*e^9)/64) + (d + e*x)^(9/2)*((5327*a^3*b^4*e^8)/128 - (5327*b^7*d^3*e^5)/128 + (15981*a*b^
6*d^2*e^6)/128 - (15981*a^2*b^5*d*e^7)/128) + (d + e*x)^(7/2)*((9443*a^4*b^3*e^9)/64 + (9443*b^7*d^4*e^5)/64 -
(9443*a*b^6*d^3*e^6)/16 - (9443*a^3*b^4*d*e^8)/16 + (28329*a^2*b^5*d^2*e^7)/32))/((d + e*x)*(5*b^13*d^4 + 5*a
^4*b^9*e^4 - 20*a^3*b^10*d*e^3 + 30*a^2*b^11*d^2*e^2 - 20*a*b^12*d^3*e) - (d + e*x)^2*(10*b^13*d^3 - 10*a^3*b^
10*e^3 + 30*a^2*b^11*d*e^2 - 30*a*b^12*d^2*e) + b^13*(d + e*x)^5 - (5*b^13*d - 5*a*b^12*e)*(d + e*x)^4 - b^13*
d^5 + (d + e*x)^3*(10*b^13*d^2 + 10*a^2*b^11*e^2 - 20*a*b^12*d*e) + a^5*b^8*e^5 - 5*a^4*b^9*d*e^4 - 10*a^2*b^1
1*d^3*e^2 + 10*a^3*b^10*d^2*e^3 + 5*a*b^12*d^4*e) + (2*e^5*(d + e*x)^(5/2))/(5*b^6) + (2*e^5*(6*b^6*d - 6*a*b^
5*e)*(d + e*x)^(3/2))/(3*b^12) - (9009*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(5/2)*(d + e*x)^(1/2))/(a^3*e^8 - b^3
*d^3*e^5 + 3*a*b^2*d^2*e^6 - 3*a^2*b*d*e^7))*(a*e - b*d)^(5/2))/(128*b^(17/2))

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