3.14.68 \(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=197 \[ -\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {693 e^5 \sqrt {d+e x}}{128 b^6} \]

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Rubi [A]  time = 0.10, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 47, 50, 63, 208} \begin {gather*} -\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {693 e^5 \sqrt {d+e x}}{128 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)) - (231*e^3*(d + e*x)^(5/2))/
(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*
x)^4) - (d + e*x)^(11/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*b^(13/2))

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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

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 63

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 208

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^{11/2}}{(a+b x)^6} \, dx\\ &=-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {(11 e) \int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (99 e^2\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2}\\ &=-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^3\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3}\\ &=-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^4\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4}\\ &=-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^5}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^6}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^4 (b d-a e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^6}\\ &=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 52, normalized size = 0.26 \begin {gather*} \frac {2 e^5 (d+e x)^{13/2} \, _2F_1\left (6,\frac {13}{2};\frac {15}{2};-\frac {b (d+e x)}{a e-b d}\right )}{13 (a e-b d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(2*e^5*(d + e*x)^(13/2)*Hypergeometric2F1[6, 13/2, 15/2, -((b*(d + e*x))/(-(b*d) + a*e))])/(13*(-(b*d) + a*e)^
6)

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IntegrateAlgebraic [B]  time = 2.48, size = 408, normalized size = 2.07 \begin {gather*} \frac {e^5 \sqrt {d+e x} \left (3465 a^5 e^5+16170 a^4 b e^4 (d+e x)-17325 a^4 b d e^4+34650 a^3 b^2 d^2 e^3+29568 a^3 b^2 e^3 (d+e x)^2-64680 a^3 b^2 d e^3 (d+e x)-34650 a^2 b^3 d^3 e^2+97020 a^2 b^3 d^2 e^2 (d+e x)+26070 a^2 b^3 e^2 (d+e x)^3-88704 a^2 b^3 d e^2 (d+e x)^2+17325 a b^4 d^4 e-64680 a b^4 d^3 e (d+e x)+88704 a b^4 d^2 e (d+e x)^2+10615 a b^4 e (d+e x)^4-52140 a b^4 d e (d+e x)^3-3465 b^5 d^5+16170 b^5 d^4 (d+e x)-29568 b^5 d^3 (d+e x)^2+26070 b^5 d^2 (d+e x)^3+1280 b^5 (d+e x)^5-10615 b^5 d (d+e x)^4\right )}{640 b^6 (a e+b (d+e x)-b d)^5}-\frac {693 \left (b d e^5-a e^6\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{13/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(e^5*Sqrt[d + e*x]*(-3465*b^5*d^5 + 17325*a*b^4*d^4*e - 34650*a^2*b^3*d^3*e^2 + 34650*a^3*b^2*d^2*e^3 - 17325*
a^4*b*d*e^4 + 3465*a^5*e^5 + 16170*b^5*d^4*(d + e*x) - 64680*a*b^4*d^3*e*(d + e*x) + 97020*a^2*b^3*d^2*e^2*(d
+ e*x) - 64680*a^3*b^2*d*e^3*(d + e*x) + 16170*a^4*b*e^4*(d + e*x) - 29568*b^5*d^3*(d + e*x)^2 + 88704*a*b^4*d
^2*e*(d + e*x)^2 - 88704*a^2*b^3*d*e^2*(d + e*x)^2 + 29568*a^3*b^2*e^3*(d + e*x)^2 + 26070*b^5*d^2*(d + e*x)^3
- 52140*a*b^4*d*e*(d + e*x)^3 + 26070*a^2*b^3*e^2*(d + e*x)^3 - 10615*b^5*d*(d + e*x)^4 + 10615*a*b^4*e*(d +
e*x)^4 + 1280*b^5*(d + e*x)^5))/(640*b^6*(-(b*d) + a*e + b*(d + e*x))^5) - (693*(b*d*e^5 - a*e^6)*ArcTan[(Sqrt
[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(128*b^(13/2)*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.43, size = 890, normalized size = 4.52 \begin {gather*} \left [\frac {3465 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (1280 \, b^{5} e^{5} x^{5} - 128 \, b^{5} d^{5} - 176 \, a b^{4} d^{4} e - 264 \, a^{2} b^{3} d^{3} e^{2} - 462 \, a^{3} b^{2} d^{2} e^{3} - 1155 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} - 5 \, {\left (843 \, b^{5} d e^{4} - 2123 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (359 \, b^{5} d^{2} e^{3} + 968 \, a b^{4} d e^{4} - 2607 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (1124 \, b^{5} d^{3} e^{2} + 2013 \, a b^{4} d^{2} e^{3} + 5247 \, a^{2} b^{3} d e^{4} - 14784 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (408 \, b^{5} d^{4} e + 616 \, a b^{4} d^{3} e^{2} + 1089 \, a^{2} b^{3} d^{2} e^{3} + 2772 \, a^{3} b^{2} d e^{4} - 8085 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{1280 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, -\frac {3465 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (1280 \, b^{5} e^{5} x^{5} - 128 \, b^{5} d^{5} - 176 \, a b^{4} d^{4} e - 264 \, a^{2} b^{3} d^{3} e^{2} - 462 \, a^{3} b^{2} d^{2} e^{3} - 1155 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} - 5 \, {\left (843 \, b^{5} d e^{4} - 2123 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (359 \, b^{5} d^{2} e^{3} + 968 \, a b^{4} d e^{4} - 2607 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (1124 \, b^{5} d^{3} e^{2} + 2013 \, a b^{4} d^{2} e^{3} + 5247 \, a^{2} b^{3} d e^{4} - 14784 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (408 \, b^{5} d^{4} e + 616 \, a b^{4} d^{3} e^{2} + 1089 \, a^{2} b^{3} d^{2} e^{3} + 2772 \, a^{3} b^{2} d e^{4} - 8085 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{640 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e
^5)*sqrt((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^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 +
3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e
^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3 + 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*
d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 + 2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^1
1*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6), -1/640*(3465*(b^5*e^5*x^5 + 5
*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(-(b*d - a*e)/b)*arcta
n(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (1280*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264
*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 + 3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*
x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3
+ 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 +
2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8
*x^2 + 5*a^4*b^7*x + a^5*b^6)]

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giac [B]  time = 0.29, size = 459, normalized size = 2.33 \begin {gather*} \frac {693 \, {\left (b d e^{5} - a e^{6}\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^{6}} + \frac {2 \, \sqrt {x e + d} e^{5}}{b^{6}} - \frac {4215 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} d e^{5} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{5} + 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{5} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt {x e + d} b^{5} d^{5} e^{5} - 4215 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{4} e^{6} + 26540 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} d e^{6} - 50304 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{6} + 39080 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt {x e + d} a b^{4} d^{4} e^{6} - 13270 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{7} + 50304 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{7} - 58620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{7} - 16768 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{8} + 39080 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{8} - 9770 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{9} + 10925 \, \sqrt {x e + d} a^{4} b d e^{9} - 2185 \, \sqrt {x e + d} a^{5} e^{10}}{640 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5} b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

693/128*(b*d*e^5 - a*e^6)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) + 2*sqrt(x*e
+ d)*e^5/b^6 - 1/640*(4215*(x*e + d)^(9/2)*b^5*d*e^5 - 13270*(x*e + d)^(7/2)*b^5*d^2*e^5 + 16768*(x*e + d)^(5
/2)*b^5*d^3*e^5 - 9770*(x*e + d)^(3/2)*b^5*d^4*e^5 + 2185*sqrt(x*e + d)*b^5*d^5*e^5 - 4215*(x*e + d)^(9/2)*a*b
^4*e^6 + 26540*(x*e + d)^(7/2)*a*b^4*d*e^6 - 50304*(x*e + d)^(5/2)*a*b^4*d^2*e^6 + 39080*(x*e + d)^(3/2)*a*b^4
*d^3*e^6 - 10925*sqrt(x*e + d)*a*b^4*d^4*e^6 - 13270*(x*e + d)^(7/2)*a^2*b^3*e^7 + 50304*(x*e + d)^(5/2)*a^2*b
^3*d*e^7 - 58620*(x*e + d)^(3/2)*a^2*b^3*d^2*e^7 + 21850*sqrt(x*e + d)*a^2*b^3*d^3*e^7 - 16768*(x*e + d)^(5/2)
*a^3*b^2*e^8 + 39080*(x*e + d)^(3/2)*a^3*b^2*d*e^8 - 21850*sqrt(x*e + d)*a^3*b^2*d^2*e^8 - 9770*(x*e + d)^(3/2
)*a^4*b*e^9 + 10925*sqrt(x*e + d)*a^4*b*d*e^9 - 2185*sqrt(x*e + d)*a^5*e^10)/(((x*e + d)*b - b*d + a*e)^5*b^6)

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maple [B]  time = 0.07, size = 673, normalized size = 3.42 \begin {gather*} \frac {437 \sqrt {e x +d}\, a^{5} e^{10}}{128 \left (b e x +a e \right )^{5} b^{6}}-\frac {2185 \sqrt {e x +d}\, a^{4} d \,e^{9}}{128 \left (b e x +a e \right )^{5} b^{5}}+\frac {2185 \sqrt {e x +d}\, a^{3} d^{2} e^{8}}{64 \left (b e x +a e \right )^{5} b^{4}}-\frac {2185 \sqrt {e x +d}\, a^{2} d^{3} e^{7}}{64 \left (b e x +a e \right )^{5} b^{3}}+\frac {2185 \sqrt {e x +d}\, a \,d^{4} e^{6}}{128 \left (b e x +a e \right )^{5} b^{2}}-\frac {437 \sqrt {e x +d}\, d^{5} e^{5}}{128 \left (b e x +a e \right )^{5} b}+\frac {977 \left (e x +d \right )^{\frac {3}{2}} a^{4} e^{9}}{64 \left (b e x +a e \right )^{5} b^{5}}-\frac {977 \left (e x +d \right )^{\frac {3}{2}} a^{3} d \,e^{8}}{16 \left (b e x +a e \right )^{5} b^{4}}+\frac {2931 \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2} e^{7}}{32 \left (b e x +a e \right )^{5} b^{3}}-\frac {977 \left (e x +d \right )^{\frac {3}{2}} a \,d^{3} e^{6}}{16 \left (b e x +a e \right )^{5} b^{2}}+\frac {977 \left (e x +d \right )^{\frac {3}{2}} d^{4} e^{5}}{64 \left (b e x +a e \right )^{5} b}+\frac {131 \left (e x +d \right )^{\frac {5}{2}} a^{3} e^{8}}{5 \left (b e x +a e \right )^{5} b^{4}}-\frac {393 \left (e x +d \right )^{\frac {5}{2}} a^{2} d \,e^{7}}{5 \left (b e x +a e \right )^{5} b^{3}}+\frac {393 \left (e x +d \right )^{\frac {5}{2}} a \,d^{2} e^{6}}{5 \left (b e x +a e \right )^{5} b^{2}}-\frac {131 \left (e x +d \right )^{\frac {5}{2}} d^{3} e^{5}}{5 \left (b e x +a e \right )^{5} b}+\frac {1327 \left (e x +d \right )^{\frac {7}{2}} a^{2} e^{7}}{64 \left (b e x +a e \right )^{5} b^{3}}-\frac {1327 \left (e x +d \right )^{\frac {7}{2}} a d \,e^{6}}{32 \left (b e x +a e \right )^{5} b^{2}}+\frac {1327 \left (e x +d \right )^{\frac {7}{2}} d^{2} e^{5}}{64 \left (b e x +a e \right )^{5} b}+\frac {843 \left (e x +d \right )^{\frac {9}{2}} a \,e^{6}}{128 \left (b e x +a e \right )^{5} b^{2}}-\frac {843 \left (e x +d \right )^{\frac {9}{2}} d \,e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {693 a \,e^{6} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{6}}+\frac {693 d \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{5}}+\frac {2 \sqrt {e x +d}\, e^{5}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

2*e^5*(e*x+d)^(1/2)/b^6+843/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(9/2)*a-843/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(9/2
)*d+1327/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a^2-1327/32*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(7/2)*a*d+1327/64*e^
5/b/(b*e*x+a*e)^5*(e*x+d)^(7/2)*d^2+131/5*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a^3-393/5*e^7/b^3/(b*e*x+a*e)^5*
(e*x+d)^(5/2)*a^2*d+393/5*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(5/2)*a*d^2-131/5*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(5/2)*d^
3+977/64*e^9/b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^4-977/16*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a^3*d+2931/32*e^7/
b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*d^2*a^2-977/16*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*a*d^3+977/64*e^5/b/(b*e*x+a
*e)^5*(e*x+d)^(3/2)*d^4+437/128*e^10/b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^5-2185/128*e^9/b^5/(b*e*x+a*e)^5*(e*x+d
)^(1/2)*a^4*d+2185/64*e^8/b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a^3*d^2-2185/64*e^7/b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*
a^2*d^3+2185/128*e^6/b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*a*d^4-437/128*e^5/b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*d^5-693/1
28*e^6/b^6/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a+693/128*e^5/b^5/((a*e-b*d)*b)^(1/
2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*d

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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)^(11/2)/(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 details)Is a*e-b*d positive or negative?

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mupad [B]  time = 0.32, size = 598, normalized size = 3.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (\frac {1327\,a^2\,b^3\,e^7}{64}-\frac {1327\,a\,b^4\,d\,e^6}{32}+\frac {1327\,b^5\,d^2\,e^5}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {437\,a^5\,e^{10}}{128}-\frac {2185\,a^4\,b\,d\,e^9}{128}+\frac {2185\,a^3\,b^2\,d^2\,e^8}{64}-\frac {2185\,a^2\,b^3\,d^3\,e^7}{64}+\frac {2185\,a\,b^4\,d^4\,e^6}{128}-\frac {437\,b^5\,d^5\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {131\,a^3\,b^2\,e^8}{5}-\frac {393\,a^2\,b^3\,d\,e^7}{5}+\frac {393\,a\,b^4\,d^2\,e^6}{5}-\frac {131\,b^5\,d^3\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {977\,a^4\,b\,e^9}{64}-\frac {977\,a^3\,b^2\,d\,e^8}{16}+\frac {2931\,a^2\,b^3\,d^2\,e^7}{32}-\frac {977\,a\,b^4\,d^3\,e^6}{16}+\frac {977\,b^5\,d^4\,e^5}{64}\right )+\left (\frac {843\,a\,b^4\,e^6}{128}-\frac {843\,b^5\,d\,e^5}{128}\right )\,{\left (d+e\,x\right )}^{9/2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b^7\,e^4-20\,a^3\,b^8\,d\,e^3+30\,a^2\,b^9\,d^2\,e^2-20\,a\,b^{10}\,d^3\,e+5\,b^{11}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^8\,e^3+30\,a^2\,b^9\,d\,e^2-30\,a\,b^{10}\,d^2\,e+10\,b^{11}\,d^3\right )+b^{11}\,{\left (d+e\,x\right )}^5-\left (5\,b^{11}\,d-5\,a\,b^{10}\,e\right )\,{\left (d+e\,x\right )}^4-b^{11}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^9\,e^2-20\,a\,b^{10}\,d\,e+10\,b^{11}\,d^2\right )+a^5\,b^6\,e^5-5\,a^4\,b^7\,d\,e^4-10\,a^2\,b^9\,d^3\,e^2+10\,a^3\,b^8\,d^2\,e^3+5\,a\,b^{10}\,d^4\,e}+\frac {2\,e^5\,\sqrt {d+e\,x}}{b^6}-\frac {693\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^6-b\,d\,e^5}\right )\,\sqrt {a\,e-b\,d}}{128\,b^{13/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^(7/2)*((1327*a^2*b^3*e^7)/64 + (1327*b^5*d^2*e^5)/64 - (1327*a*b^4*d*e^6)/32) + (d + e*x)^(1/2)*((4
37*a^5*e^10)/128 - (437*b^5*d^5*e^5)/128 + (2185*a*b^4*d^4*e^6)/128 - (2185*a^2*b^3*d^3*e^7)/64 + (2185*a^3*b^
2*d^2*e^8)/64 - (2185*a^4*b*d*e^9)/128) + (d + e*x)^(5/2)*((131*a^3*b^2*e^8)/5 - (131*b^5*d^3*e^5)/5 + (393*a*
b^4*d^2*e^6)/5 - (393*a^2*b^3*d*e^7)/5) + (d + e*x)^(3/2)*((977*a^4*b*e^9)/64 + (977*b^5*d^4*e^5)/64 - (977*a*
b^4*d^3*e^6)/16 - (977*a^3*b^2*d*e^8)/16 + (2931*a^2*b^3*d^2*e^7)/32) + ((843*a*b^4*e^6)/128 - (843*b^5*d*e^5)
/128)*(d + e*x)^(9/2))/((d + e*x)*(5*b^11*d^4 + 5*a^4*b^7*e^4 - 20*a^3*b^8*d*e^3 + 30*a^2*b^9*d^2*e^2 - 20*a*b
^10*d^3*e) - (d + e*x)^2*(10*b^11*d^3 - 10*a^3*b^8*e^3 + 30*a^2*b^9*d*e^2 - 30*a*b^10*d^2*e) + b^11*(d + e*x)^
5 - (5*b^11*d - 5*a*b^10*e)*(d + e*x)^4 - b^11*d^5 + (d + e*x)^3*(10*b^11*d^2 + 10*a^2*b^9*e^2 - 20*a*b^10*d*e
) + a^5*b^6*e^5 - 5*a^4*b^7*d*e^4 - 10*a^2*b^9*d^3*e^2 + 10*a^3*b^8*d^2*e^3 + 5*a*b^10*d^4*e) + (2*e^5*(d + e*
x)^(1/2))/b^6 - (693*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^6 - b*d*e^5))*(a*e - b*d)^(
1/2))/(128*b^(13/2))

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sympy [F(-1)]  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)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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