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

Optimal. Leaf size=262 \[ -\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^{7/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^{9/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^{3/2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {770, 21, 43} \begin {gather*} -\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}+\frac {8 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac {12 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac {8 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^5 (a+b x) (d+e x)^{7/2}}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^5 (a+b x) (d+e x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^5*(a + b*x)*(d + e*x)^(9/2)) + (8*b*(b*d - a*e)^3*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)*(d + e*x)^(7/2)) - (12*b^2*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(5*e^5*(a + b*x)*(d + e*x)^(5/2)) + (8*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)*(d
+ e*x)^(3/2)) - (2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^5*(a + b*x)*Sqrt[d + e*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^{11/2}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^4}{(d+e x)^{11/2}} \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{11/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{9/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 (d+e x)^{7/2}}-\frac {4 b^3 (b d-a e)}{e^4 (d+e x)^{5/2}}+\frac {b^4}{e^4 (d+e x)^{3/2}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x) (d+e x)^{9/2}}+\frac {8 b (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^{7/2}}-\frac {12 b^2 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^{5/2}}+\frac {8 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^{3/2}}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 172, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (35 a^4 e^4+20 a^3 b e^3 (2 d+9 e x)+6 a^2 b^2 e^2 \left (8 d^2+36 d e x+63 e^2 x^2\right )+4 a b^3 e \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4+576 d^3 e x+1008 d^2 e^2 x^2+840 d e^3 x^3+315 e^4 x^4\right )\right )}{315 e^5 (a+b x) (d+e x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(35*a^4*e^4 + 20*a^3*b*e^3*(2*d + 9*e*x) + 6*a^2*b^2*e^2*(8*d^2 + 36*d*e*x + 63*e^2*x^2)
+ 4*a*b^3*e*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + b^4*(128*d^4 + 576*d^3*e*x + 1008*d^2*e^2*x
^2 + 840*d*e^3*x^3 + 315*e^4*x^4)))/(315*e^5*(a + b*x)*(d + e*x)^(9/2))

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IntegrateAlgebraic [A]  time = 36.39, size = 241, normalized size = 0.92 \begin {gather*} -\frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (35 a^4 e^4+180 a^3 b e^3 (d+e x)-140 a^3 b d e^3+210 a^2 b^2 d^2 e^2+378 a^2 b^2 e^2 (d+e x)^2-540 a^2 b^2 d e^2 (d+e x)-140 a b^3 d^3 e+540 a b^3 d^2 e (d+e x)+420 a b^3 e (d+e x)^3-756 a b^3 d e (d+e x)^2+35 b^4 d^4-180 b^4 d^3 (d+e x)+378 b^4 d^2 (d+e x)^2+315 b^4 (d+e x)^4-420 b^4 d (d+e x)^3\right )}{315 e^4 (d+e x)^{9/2} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(a*e + b*e*x)^2/e^2]*(35*b^4*d^4 - 140*a*b^3*d^3*e + 210*a^2*b^2*d^2*e^2 - 140*a^3*b*d*e^3 + 35*a^4*e
^4 - 180*b^4*d^3*(d + e*x) + 540*a*b^3*d^2*e*(d + e*x) - 540*a^2*b^2*d*e^2*(d + e*x) + 180*a^3*b*e^3*(d + e*x)
+ 378*b^4*d^2*(d + e*x)^2 - 756*a*b^3*d*e*(d + e*x)^2 + 378*a^2*b^2*e^2*(d + e*x)^2 - 420*b^4*d*(d + e*x)^3 +
420*a*b^3*e*(d + e*x)^3 + 315*b^4*(d + e*x)^4))/(315*e^4*(d + e*x)^(9/2)*(a*e + b*e*x))

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fricas [A]  time = 0.43, size = 235, normalized size = 0.90 \begin {gather*} -\frac {2 \, {\left (315 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} + 64 \, a b^{3} d^{3} e + 48 \, a^{2} b^{2} d^{2} e^{2} + 40 \, a^{3} b d e^{3} + 35 \, a^{4} e^{4} + 420 \, {\left (2 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 126 \, {\left (8 \, b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 36 \, {\left (16 \, b^{4} d^{3} e + 8 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 5 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(315*b^4*e^4*x^4 + 128*b^4*d^4 + 64*a*b^3*d^3*e + 48*a^2*b^2*d^2*e^2 + 40*a^3*b*d*e^3 + 35*a^4*e^4 + 42
0*(2*b^4*d*e^3 + a*b^3*e^4)*x^3 + 126*(8*b^4*d^2*e^2 + 4*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 + 36*(16*b^4*d^3*e +
8*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3 + 5*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 +
10*d^3*e^7*x^2 + 5*d^4*e^6*x + d^5*e^5)

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giac [A]  time = 0.23, size = 307, normalized size = 1.17 \begin {gather*} -\frac {2 \, {\left (315 \, {\left (x e + d\right )}^{4} b^{4} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (x e + d\right )}^{3} b^{4} d \mathrm {sgn}\left (b x + a\right ) + 378 \, {\left (x e + d\right )}^{2} b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 180 \, {\left (x e + d\right )} b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (x e + d\right )}^{3} a b^{3} e \mathrm {sgn}\left (b x + a\right ) - 756 \, {\left (x e + d\right )}^{2} a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + 540 \, {\left (x e + d\right )} a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 140 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 378 \, {\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 540 \, {\left (x e + d\right )} a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 210 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (x e + d\right )} a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right ) - 140 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{315 \, {\left (x e + d\right )}^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(315*(x*e + d)^4*b^4*sgn(b*x + a) - 420*(x*e + d)^3*b^4*d*sgn(b*x + a) + 378*(x*e + d)^2*b^4*d^2*sgn(b*
x + a) - 180*(x*e + d)*b^4*d^3*sgn(b*x + a) + 35*b^4*d^4*sgn(b*x + a) + 420*(x*e + d)^3*a*b^3*e*sgn(b*x + a) -
756*(x*e + d)^2*a*b^3*d*e*sgn(b*x + a) + 540*(x*e + d)*a*b^3*d^2*e*sgn(b*x + a) - 140*a*b^3*d^3*e*sgn(b*x + a
) + 378*(x*e + d)^2*a^2*b^2*e^2*sgn(b*x + a) - 540*(x*e + d)*a^2*b^2*d*e^2*sgn(b*x + a) + 210*a^2*b^2*d^2*e^2*
sgn(b*x + a) + 180*(x*e + d)*a^3*b*e^3*sgn(b*x + a) - 140*a^3*b*d*e^3*sgn(b*x + a) + 35*a^4*e^4*sgn(b*x + a))*
e^(-5)/(x*e + d)^(9/2)

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maple [A]  time = 0.05, size = 202, normalized size = 0.77 \begin {gather*} -\frac {2 \left (315 b^{4} e^{4} x^{4}+420 a \,b^{3} e^{4} x^{3}+840 b^{4} d \,e^{3} x^{3}+378 a^{2} b^{2} e^{4} x^{2}+504 a \,b^{3} d \,e^{3} x^{2}+1008 b^{4} d^{2} e^{2} x^{2}+180 a^{3} b \,e^{4} x +216 a^{2} b^{2} d \,e^{3} x +288 a \,b^{3} d^{2} e^{2} x +576 b^{4} d^{3} e x +35 a^{4} e^{4}+40 a^{3} b d \,e^{3}+48 a^{2} b^{2} d^{2} e^{2}+64 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{315 \left (e x +d \right )^{\frac {9}{2}} \left (b x +a \right )^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/(e*x+d)^(9/2)*(315*b^4*e^4*x^4+420*a*b^3*e^4*x^3+840*b^4*d*e^3*x^3+378*a^2*b^2*e^4*x^2+504*a*b^3*d*e^3*
x^2+1008*b^4*d^2*e^2*x^2+180*a^3*b*e^4*x+216*a^2*b^2*d*e^3*x+288*a*b^3*d^2*e^2*x+576*b^4*d^3*e*x+35*a^4*e^4+40
*a^3*b*d*e^3+48*a^2*b^2*d^2*e^2+64*a*b^3*d^3*e+128*b^4*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [A]  time = 0.78, size = 371, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (105 \, b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} + 24 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} + 63 \, {\left (2 \, b^{3} d e^{2} + 3 \, a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{3} d^{2} e + 12 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} a}{315 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )} \sqrt {e x + d}} - \frac {2 \, {\left (315 \, b^{3} e^{4} x^{4} + 128 \, b^{3} d^{4} + 48 \, a b^{2} d^{3} e + 24 \, a^{2} b d^{2} e^{2} + 10 \, a^{3} d e^{3} + 105 \, {\left (8 \, b^{3} d e^{3} + 3 \, a b^{2} e^{4}\right )} x^{3} + 63 \, {\left (16 \, b^{3} d^{2} e^{2} + 6 \, a b^{2} d e^{3} + 3 \, a^{2} b e^{4}\right )} x^{2} + 9 \, {\left (64 \, b^{3} d^{3} e + 24 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d e^{3} + 5 \, a^{3} e^{4}\right )} x\right )} b}{315 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )} \sqrt {e x + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*(105*b^3*e^3*x^3 + 16*b^3*d^3 + 24*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 35*a^3*e^3 + 63*(2*b^3*d*e^2 + 3*a*b^
2*e^3)*x^2 + 9*(8*b^3*d^2*e + 12*a*b^2*d*e^2 + 15*a^2*b*e^3)*x)*a/((e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*
d^3*e^5*x + d^4*e^4)*sqrt(e*x + d)) - 2/315*(315*b^3*e^4*x^4 + 128*b^3*d^4 + 48*a*b^2*d^3*e + 24*a^2*b*d^2*e^2
+ 10*a^3*d*e^3 + 105*(8*b^3*d*e^3 + 3*a*b^2*e^4)*x^3 + 63*(16*b^3*d^2*e^2 + 6*a*b^2*d*e^3 + 3*a^2*b*e^4)*x^2
+ 9*(64*b^3*d^3*e + 24*a*b^2*d^2*e^2 + 12*a^2*b*d*e^3 + 5*a^3*e^4)*x)*b/((e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^
2 + 4*d^3*e^6*x + d^4*e^5)*sqrt(e*x + d))

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mupad [B]  time = 2.93, size = 333, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {8\,x\,\left (5\,a^3\,e^3+6\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{35\,e^8}+\frac {2\,b^3\,x^4}{e^5}+\frac {70\,a^4\,e^4+80\,a^3\,b\,d\,e^3+96\,a^2\,b^2\,d^2\,e^2+128\,a\,b^3\,d^3\,e+256\,b^4\,d^4}{315\,b\,e^9}+\frac {8\,b^2\,x^3\,\left (a\,e+2\,b\,d\right )}{3\,e^6}+\frac {4\,b\,x^2\,\left (3\,a^2\,e^2+4\,a\,b\,d\,e+8\,b^2\,d^2\right )}{5\,e^7}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (315\,a\,e^9+1260\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{315\,b\,e^9}+\frac {2\,d\,x^3\,\left (2\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^3\,x\,\left (4\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {2\,d^2\,x^2\,\left (3\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((8*x*(5*a^3*e^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(35*e^8) + (
2*b^3*x^4)/e^5 + (70*a^4*e^4 + 256*b^4*d^4 + 96*a^2*b^2*d^2*e^2 + 128*a*b^3*d^3*e + 80*a^3*b*d*e^3)/(315*b*e^9
) + (8*b^2*x^3*(a*e + 2*b*d))/(3*e^6) + (4*b*x^2*(3*a^2*e^2 + 8*b^2*d^2 + 4*a*b*d*e))/(5*e^7)))/(x^5*(d + e*x)
^(1/2) + (a*d^4*(d + e*x)^(1/2))/(b*e^4) + (x^4*(315*a*e^9 + 1260*b*d*e^8)*(d + e*x)^(1/2))/(315*b*e^9) + (2*d
*x^3*(2*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^3*x*(4*a*e + b*d)*(d + e*x)^(1/2))/(b*e^4) + (2*d^2*x^2*(3*
a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^3))

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

[Out]

Timed out

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