\(\int \frac {(d+e x)^{13/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1665]

Optimal result

Integrand size = 28, antiderivative size = 224 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}-\frac {3003 e^5 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}} \]

[Out]

1001/128*e^5*(e*x+d)^(3/2)/b^6-3003/640*e^4*(e*x+d)^(5/2)/b^5/(b*x+a)-429/320*e^3*(e*x+d)^(7/2)/b^4/(b*x+a)^2-
143/240*e^2*(e*x+d)^(9/2)/b^3/(b*x+a)^3-13/40*e*(e*x+d)^(11/2)/b^2/(b*x+a)^4-1/5*(e*x+d)^(13/2)/b/(b*x+a)^5-30
03/128*e^5*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(15/2)+3003/128*e^5*(-a*e+b*d)*(
e*x+d)^(1/2)/b^7

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3003 e^5 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}}+\frac {3003 e^5 \sqrt {d+e x} (b d-a e)}{128 b^7}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6} \]

[In]

Int[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(3003*e^5*(b*d - a*e)*Sqrt[d + e*x])/(128*b^7) + (1001*e^5*(d + e*x)^(3/2))/(128*b^6) - (3003*e^4*(d + e*x)^(5
/2))/(640*b^5*(a + b*x)) - (429*e^3*(d + e*x)^(7/2))/(320*b^4*(a + b*x)^2) - (143*e^2*(d + e*x)^(9/2))/(240*b^
3*(a + b*x)^3) - (13*e*(d + e*x)^(11/2))/(40*b^2*(a + b*x)^4) - (d + e*x)^(13/2)/(5*b*(a + b*x)^5) - (3003*e^5
*(b*d - a*e)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(15/2))

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*} \text {integral}& = \int \frac {(d+e x)^{13/2}}{(a+b x)^6} \, dx \\ & = -\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {(13 e) \int \frac {(d+e x)^{11/2}}{(a+b x)^5} \, dx}{10 b} \\ & = -\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (143 e^2\right ) \int \frac {(d+e x)^{9/2}}{(a+b x)^4} \, dx}{80 b^2} \\ & = -\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (429 e^3\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^3} \, dx}{160 b^3} \\ & = -\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^4\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{640 b^4} \\ & = -\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^5} \\ & = \frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5 (b d-a e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^6} \\ & = \frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^5 (b d-a e)^2\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^7} \\ & = \frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}+\frac {\left (3003 e^4 (b d-a e)^2\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^7} \\ & = \frac {3003 e^5 (b d-a e) \sqrt {d+e x}}{128 b^7}+\frac {1001 e^5 (d+e x)^{3/2}}{128 b^6}-\frac {3003 e^4 (d+e x)^{5/2}}{640 b^5 (a+b x)}-\frac {429 e^3 (d+e x)^{7/2}}{320 b^4 (a+b x)^2}-\frac {143 e^2 (d+e x)^{9/2}}{240 b^3 (a+b x)^3}-\frac {13 e (d+e x)^{11/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{13/2}}{5 b (a+b x)^5}-\frac {3003 e^5 (b d-a e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.57 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (45045 a^6 e^6+30030 a^5 b e^5 (-2 d+7 e x)+3003 a^4 b^2 e^4 \left (3 d^2-94 d e x+128 e^2 x^2\right )+858 a^3 b^3 e^3 \left (3 d^3+51 d^2 e x-607 d e^2 x^2+395 e^3 x^3\right )+143 a^2 b^4 e^2 \left (8 d^4+86 d^3 e x+588 d^2 e^2 x^2-3250 d e^3 x^3+965 e^4 x^4\right )+26 a b^5 e \left (24 d^5+208 d^4 e x+889 d^3 e^2 x^2+3045 d^2 e^3 x^3-7415 d e^4 x^4+640 e^5 x^5\right )+b^6 \left (384 d^6+2928 d^5 e x+10024 d^4 e^2 x^2+21070 d^3 e^3 x^3+35595 d^2 e^4 x^4-24320 d e^5 x^5-1280 e^6 x^6\right )\right )}{1920 b^7 (a+b x)^5}+\frac {3003 e^5 (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{15/2}} \]

[In]

Integrate[(d + e*x)^(13/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/1920*(Sqrt[d + e*x]*(45045*a^6*e^6 + 30030*a^5*b*e^5*(-2*d + 7*e*x) + 3003*a^4*b^2*e^4*(3*d^2 - 94*d*e*x +
128*e^2*x^2) + 858*a^3*b^3*e^3*(3*d^3 + 51*d^2*e*x - 607*d*e^2*x^2 + 395*e^3*x^3) + 143*a^2*b^4*e^2*(8*d^4 + 8
6*d^3*e*x + 588*d^2*e^2*x^2 - 3250*d*e^3*x^3 + 965*e^4*x^4) + 26*a*b^5*e*(24*d^5 + 208*d^4*e*x + 889*d^3*e^2*x
^2 + 3045*d^2*e^3*x^3 - 7415*d*e^4*x^4 + 640*e^5*x^5) + b^6*(384*d^6 + 2928*d^5*e*x + 10024*d^4*e^2*x^2 + 2107
0*d^3*e^3*x^3 + 35595*d^2*e^4*x^4 - 24320*d*e^5*x^5 - 1280*e^6*x^6)))/(b^7*(a + b*x)^5) + (3003*e^5*(-(b*d) +
a*e)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(15/2))

Maple [A] (verified)

Time = 3.41 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.30

method	result	size
risch	\(-\frac {2 e^{5} \left (-b e x +18 a e -19 b d \right ) \sqrt {e x +d}}{3 b^{7}}+\frac {\left (2 a^{2} e^{2}-4 a b d e +2 b^{2} d^{2}\right ) e^{5} \left (\frac {-\frac {2373 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}-\frac {12131 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {1253}{30} a^{2} b^{2} e^{2}+\frac {1253}{15} a \,b^{3} d e -\frac {1253}{30} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {9629}{384} a^{3} b \,e^{3}+\frac {9629}{128} a^{2} b^{2} d \,e^{2}-\frac {9629}{128} a \,b^{3} d^{2} e +\frac {9629}{384} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1467}{256} e^{4} a^{4}+\frac {1467}{64} b \,e^{3} d \,a^{3}-\frac {4401}{128} b^{2} e^{2} d^{2} a^{2}+\frac {1467}{64} a \,b^{3} d^{3} e -\frac {1467}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{7}}\)	\(292\)
pseudoelliptic	\(\frac {\frac {3003 e^{5} \left (b x +a \right )^{5} \left (a e -b d \right )^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}-\frac {3003 \left (\left (-\frac {256}{9009} e^{6} x^{6}+\frac {128}{15015} d^{6}-\frac {4864}{9009} d \,e^{5} x^{5}+\frac {113}{143} d^{2} e^{4} x^{4}+\frac {602}{1287} x^{3} d^{3} e^{3}+\frac {1432}{6435} d^{4} e^{2} x^{2}+\frac {976}{15015} d^{5} e x \right ) b^{6}+\frac {16 \left (\frac {80}{3} e^{5} x^{5}-\frac {7415}{24} x^{4} d \,e^{4}+\frac {1015}{8} d^{2} e^{3} x^{3}+\frac {889}{24} d^{3} e^{2} x^{2}+\frac {26}{3} d^{4} e x +d^{5}\right ) e a \,b^{5}}{1155}+\frac {8 \left (\frac {965}{8} e^{4} x^{4}-\frac {1625}{4} d \,e^{3} x^{3}+\frac {147}{2} d^{2} e^{2} x^{2}+\frac {43}{4} d^{3} e x +d^{4}\right ) e^{2} a^{2} b^{4}}{315}+\frac {2 e^{3} \left (\frac {395}{3} e^{3} x^{3}-\frac {607}{3} d \,e^{2} x^{2}+17 d^{2} e x +d^{3}\right ) a^{3} b^{3}}{35}+\frac {e^{4} \left (\frac {128}{3} x^{2} e^{2}-\frac {94}{3} d e x +d^{2}\right ) a^{4} b^{2}}{5}-\frac {4 \left (-\frac {7 e x}{2}+d \right ) e^{5} a^{5} b}{3}+a^{6} e^{6}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}}{128}}{\sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b^{7}}\)	\(354\)
derivativedivides	\(2 e^{5} \left (-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} b}{3}+6 a e \sqrt {e x +d}-6 d b \sqrt {e x +d}}{b^{7}}+\frac {\frac {\left (-\frac {2373}{256} a^{2} b^{4} e^{2}+\frac {2373}{128} a \,b^{5} d e -\frac {2373}{256} b^{6} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {12131 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {1253}{30} a^{4} b^{2} e^{4}+\frac {2506}{15} a^{3} b^{3} d \,e^{3}-\frac {1253}{5} a^{2} b^{4} d^{2} e^{2}+\frac {2506}{15} a \,b^{5} d^{3} e -\frac {1253}{30} d^{4} b^{6}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {9629}{384} a^{5} b \,e^{5}+\frac {48145}{384} a^{4} b^{2} d \,e^{4}-\frac {48145}{192} a^{3} b^{3} d^{2} e^{3}+\frac {48145}{192} a^{2} b^{4} d^{3} e^{2}-\frac {48145}{384} a \,b^{5} d^{4} e +\frac {9629}{384} d^{5} b^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1467}{256} a^{6} e^{6}+\frac {4401}{128} a^{5} b d \,e^{5}-\frac {22005}{256} a^{4} b^{2} d^{2} e^{4}+\frac {7335}{64} a^{3} b^{3} d^{3} e^{3}-\frac {22005}{256} a^{2} b^{4} d^{4} e^{2}+\frac {4401}{128} a \,b^{5} d^{5} e -\frac {1467}{256} b^{6} d^{6}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{7}}\right )\)	\(437\)
default	\(2 e^{5} \left (-\frac {-\frac {\left (e x +d \right )^{\frac {3}{2}} b}{3}+6 a e \sqrt {e x +d}-6 d b \sqrt {e x +d}}{b^{7}}+\frac {\frac {\left (-\frac {2373}{256} a^{2} b^{4} e^{2}+\frac {2373}{128} a \,b^{5} d e -\frac {2373}{256} b^{6} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {12131 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {7}{2}}}{384}+\left (-\frac {1253}{30} a^{4} b^{2} e^{4}+\frac {2506}{15} a^{3} b^{3} d \,e^{3}-\frac {1253}{5} a^{2} b^{4} d^{2} e^{2}+\frac {2506}{15} a \,b^{5} d^{3} e -\frac {1253}{30} d^{4} b^{6}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {9629}{384} a^{5} b \,e^{5}+\frac {48145}{384} a^{4} b^{2} d \,e^{4}-\frac {48145}{192} a^{3} b^{3} d^{2} e^{3}+\frac {48145}{192} a^{2} b^{4} d^{3} e^{2}-\frac {48145}{384} a \,b^{5} d^{4} e +\frac {9629}{384} d^{5} b^{6}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {1467}{256} a^{6} e^{6}+\frac {4401}{128} a^{5} b d \,e^{5}-\frac {22005}{256} a^{4} b^{2} d^{2} e^{4}+\frac {7335}{64} a^{3} b^{3} d^{3} e^{3}-\frac {22005}{256} a^{2} b^{4} d^{4} e^{2}+\frac {4401}{128} a \,b^{5} d^{5} e -\frac {1467}{256} b^{6} d^{6}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3003 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{7}}\right )\)	\(437\)

[In]

int((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

-2/3*e^5*(-b*e*x+18*a*e-19*b*d)*(e*x+d)^(1/2)/b^7+1/b^7*(2*a^2*e^2-4*a*b*d*e+2*b^2*d^2)*e^5*((-2373/256*b^4*(e
*x+d)^(9/2)-12131/384*(a*e-b*d)*b^3*(e*x+d)^(7/2)+(-1253/30*a^2*b^2*e^2+1253/15*a*b^3*d*e-1253/30*b^4*d^2)*(e*
x+d)^(5/2)+(-9629/384*a^3*b*e^3+9629/128*a^2*b^2*d*e^2-9629/128*a*b^3*d^2*e+9629/384*b^4*d^3)*(e*x+d)^(3/2)+(-
1467/256*e^4*a^4+1467/64*b*e^3*d*a^3-4401/128*b^2*e^2*d^2*a^2+1467/64*a*b^3*d^3*e-1467/256*b^4*d^4)*(e*x+d)^(1
/2))/(b*(e*x+d)+a*e-b*d)^5+3003/256/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (184) = 368\).

Time = 0.33 (sec) , antiderivative size = 1234, normalized size of antiderivative = 5.51 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*
(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sq
rt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(1280*b^6*e
^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*b^2*d^2*e^4 +
60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4 - 38558*a*b^5
*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^5 + 33891*a^3*
b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^3*d*e^5 + 1921
92*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*b^3*d^2*e^4 -
141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^10*x^3 + 10*a^3
*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7), -1/1920*(45045*(a^5*b*d*e^5 - a^6*e^6 + (b^6*d*e^5 - a*b^5*e^6)*x^5 + 5*(a*
b^5*d*e^5 - a^2*b^4*e^6)*x^4 + 10*(a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 10*(a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 5
*(a^4*b^2*d*e^5 - a^5*b*e^6)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e))
- (1280*b^6*e^6*x^6 - 384*b^6*d^6 - 624*a*b^5*d^5*e - 1144*a^2*b^4*d^4*e^2 - 2574*a^3*b^3*d^3*e^3 - 9009*a^4*
b^2*d^2*e^4 + 60060*a^5*b*d*e^5 - 45045*a^6*e^6 + 1280*(19*b^6*d*e^5 - 13*a*b^5*e^6)*x^5 - 5*(7119*b^6*d^2*e^4
- 38558*a*b^5*d*e^5 + 27599*a^2*b^4*e^6)*x^4 - 10*(2107*b^6*d^3*e^3 + 7917*a*b^5*d^2*e^4 - 46475*a^2*b^4*d*e^
5 + 33891*a^3*b^3*e^6)*x^3 - 2*(5012*b^6*d^4*e^2 + 11557*a*b^5*d^3*e^3 + 42042*a^2*b^4*d^2*e^4 - 260403*a^3*b^
3*d*e^5 + 192192*a^4*b^2*e^6)*x^2 - 2*(1464*b^6*d^5*e + 2704*a*b^5*d^4*e^2 + 6149*a^2*b^4*d^3*e^3 + 21879*a^3*
b^3*d^2*e^4 - 141141*a^4*b^2*d*e^5 + 105105*a^5*b*e^6)*x)*sqrt(e*x + d))/(b^12*x^5 + 5*a*b^11*x^4 + 10*a^2*b^1
0*x^3 + 10*a^3*b^9*x^2 + 5*a^4*b^8*x + a^5*b^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(13/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(13/2)/(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(a*e-b*d>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (184) = 368\).

Time = 0.30 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.73 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3003 \, {\left (b^{2} d^{2} e^{5} - 2 \, a b d e^{6} + a^{2} e^{7}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{7}} - \frac {35595 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{6} d^{2} e^{5} - 121310 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} d^{3} e^{5} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d^{4} e^{5} - 96290 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{5} e^{5} + 22005 \, \sqrt {e x + d} b^{6} d^{6} e^{5} - 71190 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{5} d e^{6} + 363930 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{5} d^{2} e^{6} - 641536 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} d^{3} e^{6} + 481450 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d^{4} e^{6} - 132030 \, \sqrt {e x + d} a b^{5} d^{5} e^{6} + 35595 \, {\left (e x + d\right )}^{\frac {9}{2}} a^{2} b^{4} e^{7} - 363930 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{4} d e^{7} + 962304 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{4} d^{2} e^{7} - 962900 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{3} e^{7} + 330075 \, \sqrt {e x + d} a^{2} b^{4} d^{4} e^{7} + 121310 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{3} b^{3} e^{8} - 641536 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{3} d e^{8} + 962900 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{3} d^{2} e^{8} - 440100 \, \sqrt {e x + d} a^{3} b^{3} d^{3} e^{8} + 160384 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{4} b^{2} e^{9} - 481450 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{2} d e^{9} + 330075 \, \sqrt {e x + d} a^{4} b^{2} d^{2} e^{9} + 96290 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{5} b e^{10} - 132030 \, \sqrt {e x + d} a^{5} b d e^{10} + 22005 \, \sqrt {e x + d} a^{6} e^{11}}{1920 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{7}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{12} e^{5} + 18 \, \sqrt {e x + d} b^{12} d e^{5} - 18 \, \sqrt {e x + d} a b^{11} e^{6}\right )}}{3 \, b^{18}} \]

[In]

integrate((e*x+d)^(13/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3003/128*(b^2*d^2*e^5 - 2*a*b*d*e^6 + a^2*e^7)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b
*e)*b^7) - 1/1920*(35595*(e*x + d)^(9/2)*b^6*d^2*e^5 - 121310*(e*x + d)^(7/2)*b^6*d^3*e^5 + 160384*(e*x + d)^(
5/2)*b^6*d^4*e^5 - 96290*(e*x + d)^(3/2)*b^6*d^5*e^5 + 22005*sqrt(e*x + d)*b^6*d^6*e^5 - 71190*(e*x + d)^(9/2)
*a*b^5*d*e^6 + 363930*(e*x + d)^(7/2)*a*b^5*d^2*e^6 - 641536*(e*x + d)^(5/2)*a*b^5*d^3*e^6 + 481450*(e*x + d)^
(3/2)*a*b^5*d^4*e^6 - 132030*sqrt(e*x + d)*a*b^5*d^5*e^6 + 35595*(e*x + d)^(9/2)*a^2*b^4*e^7 - 363930*(e*x + d
)^(7/2)*a^2*b^4*d*e^7 + 962304*(e*x + d)^(5/2)*a^2*b^4*d^2*e^7 - 962900*(e*x + d)^(3/2)*a^2*b^4*d^3*e^7 + 3300
75*sqrt(e*x + d)*a^2*b^4*d^4*e^7 + 121310*(e*x + d)^(7/2)*a^3*b^3*e^8 - 641536*(e*x + d)^(5/2)*a^3*b^3*d*e^8 +
962900*(e*x + d)^(3/2)*a^3*b^3*d^2*e^8 - 440100*sqrt(e*x + d)*a^3*b^3*d^3*e^8 + 160384*(e*x + d)^(5/2)*a^4*b^
2*e^9 - 481450*(e*x + d)^(3/2)*a^4*b^2*d*e^9 + 330075*sqrt(e*x + d)*a^4*b^2*d^2*e^9 + 96290*(e*x + d)^(3/2)*a^
5*b*e^10 - 132030*sqrt(e*x + d)*a^5*b*d*e^10 + 22005*sqrt(e*x + d)*a^6*e^11)/(((e*x + d)*b - b*d + a*e)^5*b^7)
+ 2/3*((e*x + d)^(3/2)*b^12*e^5 + 18*sqrt(e*x + d)*b^12*d*e^5 - 18*sqrt(e*x + d)*a*b^11*e^6)/b^18

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.17 \[ \int \frac {(d+e x)^{13/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2\,e^5\,{\left (d+e\,x\right )}^{3/2}}{3\,b^6}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (\frac {2373\,a^2\,b^4\,e^7}{128}-\frac {2373\,a\,b^5\,d\,e^6}{64}+\frac {2373\,b^6\,d^2\,e^5}{128}\right )+{\left (d+e\,x\right )}^{7/2}\,\left (\frac {12131\,a^3\,b^3\,e^8}{192}-\frac {12131\,a^2\,b^4\,d\,e^7}{64}+\frac {12131\,a\,b^5\,d^2\,e^6}{64}-\frac {12131\,b^6\,d^3\,e^5}{192}\right )+\sqrt {d+e\,x}\,\left (\frac {1467\,a^6\,e^{11}}{128}-\frac {4401\,a^5\,b\,d\,e^{10}}{64}+\frac {22005\,a^4\,b^2\,d^2\,e^9}{128}-\frac {7335\,a^3\,b^3\,d^3\,e^8}{32}+\frac {22005\,a^2\,b^4\,d^4\,e^7}{128}-\frac {4401\,a\,b^5\,d^5\,e^6}{64}+\frac {1467\,b^6\,d^6\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1253\,a^4\,b^2\,e^9}{15}-\frac {5012\,a^3\,b^3\,d\,e^8}{15}+\frac {2506\,a^2\,b^4\,d^2\,e^7}{5}-\frac {5012\,a\,b^5\,d^3\,e^6}{15}+\frac {1253\,b^6\,d^4\,e^5}{15}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {9629\,a^5\,b\,e^{10}}{192}-\frac {48145\,a^4\,b^2\,d\,e^9}{192}+\frac {48145\,a^3\,b^3\,d^2\,e^8}{96}-\frac {48145\,a^2\,b^4\,d^3\,e^7}{96}+\frac {48145\,a\,b^5\,d^4\,e^6}{192}-\frac {9629\,b^6\,d^5\,e^5}{192}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^8\,e^4-20\,a^3\,b^9\,d\,e^3+30\,a^2\,b^{10}\,d^2\,e^2-20\,a\,b^{11}\,d^3\,e+5\,b^{12}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^9\,e^3+30\,a^2\,b^{10}\,d\,e^2-30\,a\,b^{11}\,d^2\,e+10\,b^{12}\,d^3\right )+b^{12}\,{\left (d+e\,x\right )}^5-\left (5\,b^{12}\,d-5\,a\,b^{11}\,e\right )\,{\left (d+e\,x\right )}^4-b^{12}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{10}\,e^2-20\,a\,b^{11}\,d\,e+10\,b^{12}\,d^2\right )+a^5\,b^7\,e^5-5\,a^4\,b^8\,d\,e^4-10\,a^2\,b^{10}\,d^3\,e^2+10\,a^3\,b^9\,d^2\,e^3+5\,a\,b^{11}\,d^4\,e}+\frac {2\,e^5\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )\,\sqrt {d+e\,x}}{b^{12}}+\frac {3003\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^7-2\,a\,b\,d\,e^6+b^2\,d^2\,e^5}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{128\,b^{15/2}} \]

[In]

int((d + e*x)^(13/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(2*e^5*(d + e*x)^(3/2))/(3*b^6) - ((d + e*x)^(9/2)*((2373*a^2*b^4*e^7)/128 + (2373*b^6*d^2*e^5)/128 - (2373*a*
b^5*d*e^6)/64) + (d + e*x)^(7/2)*((12131*a^3*b^3*e^8)/192 - (12131*b^6*d^3*e^5)/192 + (12131*a*b^5*d^2*e^6)/64
- (12131*a^2*b^4*d*e^7)/64) + (d + e*x)^(1/2)*((1467*a^6*e^11)/128 + (1467*b^6*d^6*e^5)/128 - (4401*a*b^5*d^5
*e^6)/64 + (22005*a^2*b^4*d^4*e^7)/128 - (7335*a^3*b^3*d^3*e^8)/32 + (22005*a^4*b^2*d^2*e^9)/128 - (4401*a^5*b
*d*e^10)/64) + (d + e*x)^(5/2)*((1253*a^4*b^2*e^9)/15 + (1253*b^6*d^4*e^5)/15 - (5012*a*b^5*d^3*e^6)/15 - (501
2*a^3*b^3*d*e^8)/15 + (2506*a^2*b^4*d^2*e^7)/5) + (d + e*x)^(3/2)*((9629*a^5*b*e^10)/192 - (9629*b^6*d^5*e^5)/
192 + (48145*a*b^5*d^4*e^6)/192 - (48145*a^4*b^2*d*e^9)/192 - (48145*a^2*b^4*d^3*e^7)/96 + (48145*a^3*b^3*d^2*
e^8)/96))/((d + e*x)*(5*b^12*d^4 + 5*a^4*b^8*e^4 - 20*a^3*b^9*d*e^3 + 30*a^2*b^10*d^2*e^2 - 20*a*b^11*d^3*e) -
(d + e*x)^2*(10*b^12*d^3 - 10*a^3*b^9*e^3 + 30*a^2*b^10*d*e^2 - 30*a*b^11*d^2*e) + b^12*(d + e*x)^5 - (5*b^12
*d - 5*a*b^11*e)*(d + e*x)^4 - b^12*d^5 + (d + e*x)^3*(10*b^12*d^2 + 10*a^2*b^10*e^2 - 20*a*b^11*d*e) + a^5*b^
7*e^5 - 5*a^4*b^8*d*e^4 - 10*a^2*b^10*d^3*e^2 + 10*a^3*b^9*d^2*e^3 + 5*a*b^11*d^4*e) + (2*e^5*(6*b^6*d - 6*a*b
^5*e)*(d + e*x)^(1/2))/b^12 + (3003*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(3/2)*(d + e*x)^(1/2))/(a^2*e^7 + b^2*d^
2*e^5 - 2*a*b*d*e^6))*(a*e - b*d)^(3/2))/(128*b^(15/2))