3.15.2 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^6 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt {d+e x}}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}} \]

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Rubi [A]  time = 0.10, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {646, 43} \begin {gather*} \frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x}}{e^6 (a+b x)}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^6 (a+b x) \sqrt {d+e x}}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {4 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{7 e^6 (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)^5}{9 e^6 (a+b x) (d+e x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) - (10*b*(b*d - a*e)^4*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) + (4*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(e^6*(a + b*x)*(d + e*x)^(5/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d
+ e*x)^(3/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*x]) + (2*b^5*Sqrt
[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(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 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{(d+e x)^{11/2}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5}{e^5 (d+e x)^{11/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{9/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{7/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{5/2}}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^{3/2}}+\frac {b^{10}}{e^5 \sqrt {d+e x}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x) (d+e x)^{7/2}}+\frac {4 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}+\frac {10 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}+\frac {2 b^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 235, normalized size = 0.75 \begin {gather*} -\frac {2 \sqrt {(a+b x)^2} \left (7 a^5 e^5+5 a^4 b e^4 (2 d+9 e x)+2 a^3 b^2 e^3 \left (8 d^2+36 d e x+63 e^2 x^2\right )+2 a^2 b^3 e^2 \left (16 d^3+72 d^2 e x+126 d e^2 x^2+105 e^3 x^3\right )+a b^4 e \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 )-\left (b^5 \left (256 d^5+1152 d^4 e x+2016 d^3 e^2 x^2+1680 d^2 e^3 x^3+630 d e^4 x^4+63 e^5 x^5\right )\right )\right )}{63 e^6 (a+b x) (d+e x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(7*a^5*e^5 + 5*a^4*b*e^4*(2*d + 9*e*x) + 2*a^3*b^2*e^3*(8*d^2 + 36*d*e*x + 63*e^2*x^2) +
2*a^2*b^3*e^2*(16*d^3 + 72*d^2*e*x + 126*d*e^2*x^2 + 105*e^3*x^3) + a*b^4*e*(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) - b^5*(256*d^5 + 1152*d^4*e*x + 2016*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 +
630*d*e^4*x^4 + 63*e^5*x^5)))/(63*e^6*(a + b*x)*(d + e*x)^(9/2))

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IntegrateAlgebraic [A]  time = 26.05, size = 343, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-7 a^5 e^5-45 a^4 b e^4 (d+e x)+35 a^4 b d e^4-70 a^3 b^2 d^2 e^3-126 a^3 b^2 e^3 (d+e x)^2+180 a^3 b^2 d e^3 (d+e x)+70 a^2 b^3 d^3 e^2-270 a^2 b^3 d^2 e^2 (d+e x)-210 a^2 b^3 e^2 (d+e x)^3+378 a^2 b^3 d e^2 (d+e x)^2-35 a b^4 d^4 e+180 a b^4 d^3 e (d+e x)-378 a b^4 d^2 e (d+e x)^2-315 a b^4 e (d+e x)^4+420 a b^4 d e (d+e x)^3+7 b^5 d^5-45 b^5 d^4 (d+e x)+126 b^5 d^3 (d+e x)^2-210 b^5 d^2 (d+e x)^3+63 b^5 (d+e x)^5+315 b^5 d (d+e x)^4\right )}{63 e^5 (d+e x)^{9/2} (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(7*b^5*d^5 - 35*a*b^4*d^4*e + 70*a^2*b^3*d^3*e^2 - 70*a^3*b^2*d^2*e^3 + 35*a^4*b*
d*e^4 - 7*a^5*e^5 - 45*b^5*d^4*(d + e*x) + 180*a*b^4*d^3*e*(d + e*x) - 270*a^2*b^3*d^2*e^2*(d + e*x) + 180*a^3
*b^2*d*e^3*(d + e*x) - 45*a^4*b*e^4*(d + e*x) + 126*b^5*d^3*(d + e*x)^2 - 378*a*b^4*d^2*e*(d + e*x)^2 + 378*a^
2*b^3*d*e^2*(d + e*x)^2 - 126*a^3*b^2*e^3*(d + e*x)^2 - 210*b^5*d^2*(d + e*x)^3 + 420*a*b^4*d*e*(d + e*x)^3 -
210*a^2*b^3*e^2*(d + e*x)^3 + 315*b^5*d*(d + e*x)^4 - 315*a*b^4*e*(d + e*x)^4 + 63*b^5*(d + e*x)^5))/(63*e^5*(
d + e*x)^(9/2)*(a*e + b*e*x))

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fricas [A]  time = 0.40, size = 316, normalized size = 1.01 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 12
6*(16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b^4*d^3*e^2
- 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*
x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)

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giac [A]  time = 0.26, size = 450, normalized size = 1.43 \begin {gather*} 2 \, \sqrt {x e + d} b^{5} e^{\left (-6\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {2 \, {\left (315 \, {\left (x e + d\right )}^{4} b^{5} d \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) - 45 \, {\left (x e + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 315 \, {\left (x e + d\right )}^{4} a b^{4} e \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (x e + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) - 378 \, {\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (x e + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 35 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (x e + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 378 \, {\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 270 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 126 \, {\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 180 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 70 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 45 \, {\left (x e + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - 7 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{63 \, {\left (x e + d\right )}^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*b^5*e^(-6)*sgn(b*x + a) + 2/63*(315*(x*e + d)^4*b^5*d*sgn(b*x + a) - 210*(x*e + d)^3*b^5*d^2*s
gn(b*x + a) + 126*(x*e + d)^2*b^5*d^3*sgn(b*x + a) - 45*(x*e + d)*b^5*d^4*sgn(b*x + a) + 7*b^5*d^5*sgn(b*x + a
) - 315*(x*e + d)^4*a*b^4*e*sgn(b*x + a) + 420*(x*e + d)^3*a*b^4*d*e*sgn(b*x + a) - 378*(x*e + d)^2*a*b^4*d^2*
e*sgn(b*x + a) + 180*(x*e + d)*a*b^4*d^3*e*sgn(b*x + a) - 35*a*b^4*d^4*e*sgn(b*x + a) - 210*(x*e + d)^3*a^2*b^
3*e^2*sgn(b*x + a) + 378*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) - 270*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x + a) +
70*a^2*b^3*d^3*e^2*sgn(b*x + a) - 126*(x*e + d)^2*a^3*b^2*e^3*sgn(b*x + a) + 180*(x*e + d)*a^3*b^2*d*e^3*sgn(
b*x + a) - 70*a^3*b^2*d^2*e^3*sgn(b*x + a) - 45*(x*e + d)*a^4*b*e^4*sgn(b*x + a) + 35*a^4*b*d*e^4*sgn(b*x + a)
- 7*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(9/2)

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maple [A]  time = 0.05, size = 289, normalized size = 0.92 \begin {gather*} -\frac {2 \left (-63 b^{5} e^{5} x^{5}+315 a \,b^{4} e^{5} x^{4}-630 b^{5} d \,e^{4} x^{4}+210 a^{2} b^{3} e^{5} x^{3}+840 a \,b^{4} d \,e^{4} x^{3}-1680 b^{5} d^{2} e^{3} x^{3}+126 a^{3} b^{2} e^{5} x^{2}+252 a^{2} b^{3} d \,e^{4} x^{2}+1008 a \,b^{4} d^{2} e^{3} x^{2}-2016 b^{5} d^{3} e^{2} x^{2}+45 a^{4} b \,e^{5} x +72 a^{3} b^{2} d \,e^{4} x +144 a^{2} b^{3} d^{2} e^{3} x +576 a \,b^{4} d^{3} e^{2} x -1152 b^{5} d^{4} e x +7 a^{5} e^{5}+10 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+32 a^{2} b^{3} d^{3} e^{2}+128 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {9}{2}} \left (b x +a \right )^{5} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x)

[Out]

-2/63/(e*x+d)^(9/2)*(-63*b^5*e^5*x^5+315*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+210*a^2*b^3*e^5*x^3+840*a*b^4*d*e^4*x
^3-1680*b^5*d^2*e^3*x^3+126*a^3*b^2*e^5*x^2+252*a^2*b^3*d*e^4*x^2+1008*a*b^4*d^2*e^3*x^2-2016*b^5*d^3*e^2*x^2+
45*a^4*b*e^5*x+72*a^3*b^2*d*e^4*x+144*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-1152*b^5*d^4*e*x+7*a^5*e^5+10*a^4*
b*d*e^4+16*a^3*b^2*d^2*e^3+32*a^2*b^3*d^3*e^2+128*a*b^4*d^4*e-256*b^5*d^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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maxima [A]  time = 1.34, size = 305, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \, {\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \, {\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \, {\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \, {\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )}}{63 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt {e x + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(11/2),x, algorithm="maxima")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^
4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4 + 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 12
6*(16*b^5*d^3*e^2 - 8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b^4*d^3*e^2
- 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)/((e^10*x^4 + 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7
*x + d^4*e^6)*sqrt(e*x + d))

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mupad [B]  time = 1.69, size = 416, normalized size = 1.32 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {14\,a^5\,e^5+20\,a^4\,b\,d\,e^4+32\,a^3\,b^2\,d^2\,e^3+64\,a^2\,b^3\,d^3\,e^2+256\,a\,b^4\,d^4\,e-512\,b^5\,d^5}{63\,b\,e^{10}}-\frac {2\,b^4\,x^5}{e^5}+\frac {10\,b^3\,x^4\,\left (a\,e-2\,b\,d\right )}{e^6}+\frac {x\,\left (90\,a^4\,b\,e^5+144\,a^3\,b^2\,d\,e^4+288\,a^2\,b^3\,d^2\,e^3+1152\,a\,b^4\,d^3\,e^2-2304\,b^5\,d^4\,e\right )}{63\,b\,e^{10}}+\frac {20\,b^2\,x^3\,\left (a^2\,e^2+4\,a\,b\,d\,e-8\,b^2\,d^2\right )}{3\,e^7}+\frac {4\,b\,x^2\,\left (a^3\,e^3+2\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{e^8}\right )}{x^5\,\sqrt {d+e\,x}+\frac {a\,d^4\,\sqrt {d+e\,x}}{b\,e^4}+\frac {x^4\,\left (63\,a\,e^{10}+252\,b\,d\,e^9\right )\,\sqrt {d+e\,x}}{63\,b\,e^{10}}+\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^2 + b^2*x^2 + 2*a*b*x)^(5/2)/(d + e*x)^(11/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((14*a^5*e^5 - 512*b^5*d^5 + 64*a^2*b^3*d^3*e^2 + 32*a^3*b^2*d^2*e^3 + 256*a
*b^4*d^4*e + 20*a^4*b*d*e^4)/(63*b*e^10) - (2*b^4*x^5)/e^5 + (10*b^3*x^4*(a*e - 2*b*d))/e^6 + (x*(90*a^4*b*e^5
- 2304*b^5*d^4*e + 1152*a*b^4*d^3*e^2 + 144*a^3*b^2*d*e^4 + 288*a^2*b^3*d^2*e^3))/(63*b*e^10) + (20*b^2*x^3*(
a^2*e^2 - 8*b^2*d^2 + 4*a*b*d*e))/(3*e^7) + (4*b*x^2*(a^3*e^3 - 16*b^3*d^3 + 8*a*b^2*d^2*e + 2*a^2*b*d*e^2))/e
^8))/(x^5*(d + e*x)^(1/2) + (a*d^4*(d + e*x)^(1/2))/(b*e^4) + (x^4*(63*a*e^10 + 252*b*d*e^9)*(d + e*x)^(1/2))/
(63*b*e^10) + (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**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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