3.1701 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=316 \[ \frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}} \]

[Out]

2/11*(-a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(11/2)-10/9*b*(-a*e+b*d)^4*((b*x+a)^2)^(1/2)/e^6/(b*x+
a)/(e*x+d)^(9/2)+20/7*b^2*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(7/2)-4*b^3*(-a*e+b*d)^2*((b*x+a)
^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(5/2)+10/3*b^4*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(3/2)-2*b^5*((b*
x+a)^2)^(1/2)/e^6/(b*x+a)/(e*x+d)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 316, 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} \[ -\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}+\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{11 e^6 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)*(d + e*x)^(11/2)) - (10*b*(b*d - a*e)^4*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)*(d + e*x)^(9/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*
x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) - (4*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(
d + e*x)^(5/2)) + (10*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)*(d + e*x)^(3/2)) - (2*b^
5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[d + e*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)^{13/2}} \, 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)^{13/2}} \, 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)^{13/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{11/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{9/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{7/2}}-\frac {5 b^9 (b d-a e)}{e^5 (d+e x)^{5/2}}+\frac {b^{10}}{e^5 (d+e x)^{3/2}}\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}}{11 e^6 (a+b x) (d+e x)^{11/2}}-\frac {10 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x) (d+e x)^{9/2}}+\frac {20 b^2 (b d-a e)^3 \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^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {10 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 234, normalized size = 0.74 \[ -\frac {2 \sqrt {(a+b x)^2} \left (63 a^5 e^5+35 a^4 b e^4 (2 d+11 e x)+10 a^3 b^2 e^3 \left (8 d^2+44 d e x+99 e^2 x^2\right )+6 a^2 b^3 e^2 \left (16 d^3+88 d^2 e x+198 d e^2 x^2+231 e^3 x^3\right )+a b^4 e \left (128 d^4+704 d^3 e x+1584 d^2 e^2 x^2+1848 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (256 d^5+1408 d^4 e x+3168 d^3 e^2 x^2+3696 d^2 e^3 x^3+2310 d e^4 x^4+693 e^5 x^5\right )\right )}{693 e^6 (a+b x) (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(13/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(63*a^5*e^5 + 35*a^4*b*e^4*(2*d + 11*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 44*d*e*x + 99*e^2*x^
2) + 6*a^2*b^3*e^2*(16*d^3 + 88*d^2*e*x + 198*d*e^2*x^2 + 231*e^3*x^3) + a*b^4*e*(128*d^4 + 704*d^3*e*x + 1584
*d^2*e^2*x^2 + 1848*d*e^3*x^3 + 1155*e^4*x^4) + b^5*(256*d^5 + 1408*d^4*e*x + 3168*d^3*e^2*x^2 + 3696*d^2*e^3*
x^3 + 2310*d*e^4*x^4 + 693*e^5*x^5)))/(693*e^6*(a + b*x)*(d + e*x)^(11/2))

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fricas [A]  time = 0.63, size = 326, normalized size = 1.03 \[ -\frac {2 \, {\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \, {\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \, {\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \, {\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \, {\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]

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)^(13/2),x, algorithm="fricas")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^12*x^6 + 6*d*e^11*x^5 +
15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)

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giac [A]  time = 0.29, size = 447, normalized size = 1.41 \[ -\frac {2 \, {\left (693 \, {\left (x e + d\right )}^{5} b^{5} \mathrm {sgn}\left (b x + a\right ) - 1155 \, {\left (x e + d\right )}^{4} b^{5} d \mathrm {sgn}\left (b x + a\right ) + 1386 \, {\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 990 \, {\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) + 385 \, {\left (x e + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 63 \, b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 1155 \, {\left (x e + d\right )}^{4} a b^{4} e \mathrm {sgn}\left (b x + a\right ) - 2772 \, {\left (x e + d\right )}^{3} a b^{4} d e \mathrm {sgn}\left (b x + a\right ) + 2970 \, {\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 1540 \, {\left (x e + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 315 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 1386 \, {\left (x e + d\right )}^{3} a^{2} b^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 2970 \, {\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 2310 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 630 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 990 \, {\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 1540 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 630 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 385 \, {\left (x e + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 315 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 63 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{693 \, {\left (x e + d\right )}^{\frac {11}{2}}} \]

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)^(13/2),x, algorithm="giac")

[Out]

-2/693*(693*(x*e + d)^5*b^5*sgn(b*x + a) - 1155*(x*e + d)^4*b^5*d*sgn(b*x + a) + 1386*(x*e + d)^3*b^5*d^2*sgn(
b*x + a) - 990*(x*e + d)^2*b^5*d^3*sgn(b*x + a) + 385*(x*e + d)*b^5*d^4*sgn(b*x + a) - 63*b^5*d^5*sgn(b*x + a)
+ 1155*(x*e + d)^4*a*b^4*e*sgn(b*x + a) - 2772*(x*e + d)^3*a*b^4*d*e*sgn(b*x + a) + 2970*(x*e + d)^2*a*b^4*d^
2*e*sgn(b*x + a) - 1540*(x*e + d)*a*b^4*d^3*e*sgn(b*x + a) + 315*a*b^4*d^4*e*sgn(b*x + a) + 1386*(x*e + d)^3*a
^2*b^3*e^2*sgn(b*x + a) - 2970*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) + 2310*(x*e + d)*a^2*b^3*d^2*e^2*sgn(b*x
+ a) - 630*a^2*b^3*d^3*e^2*sgn(b*x + a) + 990*(x*e + d)^2*a^3*b^2*e^3*sgn(b*x + a) - 1540*(x*e + d)*a^3*b^2*d
*e^3*sgn(b*x + a) + 630*a^3*b^2*d^2*e^3*sgn(b*x + a) + 385*(x*e + d)*a^4*b*e^4*sgn(b*x + a) - 315*a^4*b*d*e^4*
sgn(b*x + a) + 63*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(11/2)

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maple [A]  time = 0.05, size = 289, normalized size = 0.91 \[ -\frac {2 \left (693 b^{5} e^{5} x^{5}+1155 a \,b^{4} e^{5} x^{4}+2310 b^{5} d \,e^{4} x^{4}+1386 a^{2} b^{3} e^{5} x^{3}+1848 a \,b^{4} d \,e^{4} x^{3}+3696 b^{5} d^{2} e^{3} x^{3}+990 a^{3} b^{2} e^{5} x^{2}+1188 a^{2} b^{3} d \,e^{4} x^{2}+1584 a \,b^{4} d^{2} e^{3} x^{2}+3168 b^{5} d^{3} e^{2} x^{2}+385 a^{4} b \,e^{5} x +440 a^{3} b^{2} d \,e^{4} x +528 a^{2} b^{3} d^{2} e^{3} x +704 a \,b^{4} d^{3} e^{2} x +1408 b^{5} d^{4} e x +63 a^{5} e^{5}+70 a^{4} b d \,e^{4}+80 a^{3} b^{2} d^{2} e^{3}+96 a^{2} b^{3} d^{3} e^{2}+128 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{\frac {11}{2}} \left (b x +a \right )^{5} e^{6}} \]

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)^(13/2),x)

[Out]

-2/693/(e*x+d)^(11/2)*(693*b^5*e^5*x^5+1155*a*b^4*e^5*x^4+2310*b^5*d*e^4*x^4+1386*a^2*b^3*e^5*x^3+1848*a*b^4*d
*e^4*x^3+3696*b^5*d^2*e^3*x^3+990*a^3*b^2*e^5*x^2+1188*a^2*b^3*d*e^4*x^2+1584*a*b^4*d^2*e^3*x^2+3168*b^5*d^3*e
^2*x^2+385*a^4*b*e^5*x+440*a^3*b^2*d*e^4*x+528*a^2*b^3*d^2*e^3*x+704*a*b^4*d^3*e^2*x+1408*b^5*d^4*e*x+63*a^5*e
^5+70*a^4*b*d*e^4+80*a^3*b^2*d^2*e^3+96*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 = 1.29, size = 315, normalized size = 1.00 \[ -\frac {2 \, {\left (693 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 128 \, a b^{4} d^{4} e + 96 \, a^{2} b^{3} d^{3} e^{2} + 80 \, a^{3} b^{2} d^{2} e^{3} + 70 \, a^{4} b d e^{4} + 63 \, a^{5} e^{5} + 1155 \, {\left (2 \, b^{5} d e^{4} + a b^{4} e^{5}\right )} x^{4} + 462 \, {\left (8 \, b^{5} d^{2} e^{3} + 4 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 198 \, {\left (16 \, b^{5} d^{3} e^{2} + 8 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 11 \, {\left (128 \, b^{5} d^{4} e + 64 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 40 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )}}{693 \, {\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )} \sqrt {e x + d}} \]

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)^(13/2),x, algorithm="maxima")

[Out]

-2/693*(693*b^5*e^5*x^5 + 256*b^5*d^5 + 128*a*b^4*d^4*e + 96*a^2*b^3*d^3*e^2 + 80*a^3*b^2*d^2*e^3 + 70*a^4*b*d
*e^4 + 63*a^5*e^5 + 1155*(2*b^5*d*e^4 + a*b^4*e^5)*x^4 + 462*(8*b^5*d^2*e^3 + 4*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x
^3 + 198*(16*b^5*d^3*e^2 + 8*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + 5*a^3*b^2*e^5)*x^2 + 11*(128*b^5*d^4*e + 64*a*b
^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 40*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)/((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)*sqrt(e*x + d))

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mupad [B]  time = 1.73, size = 437, normalized size = 1.38 \[ -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {126\,a^5\,e^5+140\,a^4\,b\,d\,e^4+160\,a^3\,b^2\,d^2\,e^3+192\,a^2\,b^3\,d^3\,e^2+256\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{693\,b\,e^{11}}+\frac {2\,b^4\,x^5}{e^6}+\frac {10\,b^3\,x^4\,\left (a\,e+2\,b\,d\right )}{3\,e^7}+\frac {x\,\left (770\,a^4\,b\,e^5+880\,a^3\,b^2\,d\,e^4+1056\,a^2\,b^3\,d^2\,e^3+1408\,a\,b^4\,d^3\,e^2+2816\,b^5\,d^4\,e\right )}{693\,b\,e^{11}}+\frac {4\,b^2\,x^3\,\left (3\,a^2\,e^2+4\,a\,b\,d\,e+8\,b^2\,d^2\right )}{3\,e^8}+\frac {4\,b\,x^2\,\left (5\,a^3\,e^3+6\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{7\,e^9}\right )}{x^6\,\sqrt {d+e\,x}+\frac {a\,d^5\,\sqrt {d+e\,x}}{b\,e^5}+\frac {x^5\,\left (a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {5\,d\,x^4\,\left (a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^4\,x\,\left (5\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}+\frac {10\,d^2\,x^3\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}} \]

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)^(13/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((126*a^5*e^5 + 512*b^5*d^5 + 192*a^2*b^3*d^3*e^2 + 160*a^3*b^2*d^2*e^3 + 25
6*a*b^4*d^4*e + 140*a^4*b*d*e^4)/(693*b*e^11) + (2*b^4*x^5)/e^6 + (10*b^3*x^4*(a*e + 2*b*d))/(3*e^7) + (x*(770
*a^4*b*e^5 + 2816*b^5*d^4*e + 1408*a*b^4*d^3*e^2 + 880*a^3*b^2*d*e^4 + 1056*a^2*b^3*d^2*e^3))/(693*b*e^11) + (
4*b^2*x^3*(3*a^2*e^2 + 8*b^2*d^2 + 4*a*b*d*e))/(3*e^8) + (4*b*x^2*(5*a^3*e^3 + 16*b^3*d^3 + 8*a*b^2*d^2*e + 6*
a^2*b*d*e^2))/(7*e^9)))/(x^6*(d + e*x)^(1/2) + (a*d^5*(d + e*x)^(1/2))/(b*e^5) + (x^5*(a*e + 5*b*d)*(d + e*x)^
(1/2))/(b*e) + (5*d*x^4*(a*e + 2*b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^4*x*(5*a*e + b*d)*(d + e*x)^(1/2))/(b*e^5)
+ (10*d^2*x^3*(a*e + b*d)*(d + e*x)^(1/2))/(b*e^3) + (5*d^3*x^2*(2*a*e + b*d)*(d + e*x)^(1/2))/(b*e^4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(13/2),x)

[Out]

Timed out

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