3.14.98 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

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

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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} (d+e x)^{9/2}}{9 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^6 (a+b x)}+\frac {4 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^6 (a+b x)}-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)^4}{e^6 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^6 (a+b x) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.13, size = 232, normalized size = 0.74 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-63 a^5 e^5+315 a^4 b e^4 (2 d+e x)+210 a^3 b^2 e^3 \left (-8 d^2-4 d e x+e^2 x^2\right )+126 a^2 b^3 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+9 a b^4 e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+b^5 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )}{63 e^6 (a+b x) \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 25.27, size = 343, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-63 a^5 e^5+315 a^4 b e^4 (d+e x)+315 a^4 b d e^4-630 a^3 b^2 d^2 e^3+210 a^3 b^2 e^3 (d+e x)^2-1260 a^3 b^2 d e^3 (d+e x)+630 a^2 b^3 d^3 e^2+1890 a^2 b^3 d^2 e^2 (d+e x)+126 a^2 b^3 e^2 (d+e x)^3-630 a^2 b^3 d e^2 (d+e x)^2-315 a b^4 d^4 e-1260 a b^4 d^3 e (d+e x)+630 a b^4 d^2 e (d+e x)^2+45 a b^4 e (d+e x)^4-252 a b^4 d e (d+e x)^3+63 b^5 d^5+315 b^5 d^4 (d+e x)-210 b^5 d^3 (d+e x)^2+126 b^5 d^2 (d+e x)^3+7 b^5 (d+e x)^5-45 b^5 d (d+e x)^4\right )}{63 e^5 \sqrt {d+e x} (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)^(3/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(63*b^5*d^5 - 315*a*b^4*d^4*e + 630*a^2*b^3*d^3*e^2 - 630*a^3*b^2*d^2*e^3 + 315*a
^4*b*d*e^4 - 63*a^5*e^5 + 315*b^5*d^4*(d + e*x) - 1260*a*b^4*d^3*e*(d + e*x) + 1890*a^2*b^3*d^2*e^2*(d + e*x)
- 1260*a^3*b^2*d*e^3*(d + e*x) + 315*a^4*b*e^4*(d + e*x) - 210*b^5*d^3*(d + e*x)^2 + 630*a*b^4*d^2*e*(d + e*x)
^2 - 630*a^2*b^3*d*e^2*(d + e*x)^2 + 210*a^3*b^2*e^3*(d + e*x)^2 + 126*b^5*d^2*(d + e*x)^3 - 252*a*b^4*d*e*(d
+ e*x)^3 + 126*a^2*b^3*e^2*(d + e*x)^3 - 45*b^5*d*(d + e*x)^4 + 45*a*b^4*e*(d + e*x)^4 + 7*b^5*(d + e*x)^5))/(
63*e^5*Sqrt[d + e*x]*(a*e + b*e*x))

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fricas [A]  time = 0.40, size = 271, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{63 \, {\left (e^{7} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^7*x + d*e^6)

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giac [B]  time = 0.23, size = 472, normalized size = 1.50 \begin {gather*} \frac {2}{63} \, {\left (7 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{5} e^{48} \mathrm {sgn}\left (b x + a\right ) - 45 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} d e^{48} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{48} \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{48} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} b^{5} d^{4} e^{48} \mathrm {sgn}\left (b x + a\right ) + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{4} e^{49} \mathrm {sgn}\left (b x + a\right ) - 252 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d e^{49} \mathrm {sgn}\left (b x + a\right ) + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{49} \mathrm {sgn}\left (b x + a\right ) - 1260 \, \sqrt {x e + d} a b^{4} d^{3} e^{49} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{50} \mathrm {sgn}\left (b x + a\right ) - 630 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{50} \mathrm {sgn}\left (b x + a\right ) + 1890 \, \sqrt {x e + d} a^{2} b^{3} d^{2} e^{50} \mathrm {sgn}\left (b x + a\right ) + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{51} \mathrm {sgn}\left (b x + a\right ) - 1260 \, \sqrt {x e + d} a^{3} b^{2} d e^{51} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {x e + d} a^{4} b e^{52} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-54\right )} + \frac {2 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{\sqrt {x e + 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)^(3/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^5*e^48*sgn(b*x + a) - 45*(x*e + d)^(7/2)*b^5*d*e^48*sgn(b*x + a) + 126*(x*e + d)^(5/
2)*b^5*d^2*e^48*sgn(b*x + a) - 210*(x*e + d)^(3/2)*b^5*d^3*e^48*sgn(b*x + a) + 315*sqrt(x*e + d)*b^5*d^4*e^48*
sgn(b*x + a) + 45*(x*e + d)^(7/2)*a*b^4*e^49*sgn(b*x + a) - 252*(x*e + d)^(5/2)*a*b^4*d*e^49*sgn(b*x + a) + 63
0*(x*e + d)^(3/2)*a*b^4*d^2*e^49*sgn(b*x + a) - 1260*sqrt(x*e + d)*a*b^4*d^3*e^49*sgn(b*x + a) + 126*(x*e + d)
^(5/2)*a^2*b^3*e^50*sgn(b*x + a) - 630*(x*e + d)^(3/2)*a^2*b^3*d*e^50*sgn(b*x + a) + 1890*sqrt(x*e + d)*a^2*b^
3*d^2*e^50*sgn(b*x + a) + 210*(x*e + d)^(3/2)*a^3*b^2*e^51*sgn(b*x + a) - 1260*sqrt(x*e + d)*a^3*b^2*d*e^51*sg
n(b*x + a) + 315*sqrt(x*e + d)*a^4*b*e^52*sgn(b*x + a))*e^(-54) + 2*(b^5*d^5*sgn(b*x + a) - 5*a*b^4*d^4*e*sgn(
b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^
5*e^5*sgn(b*x + a))*e^(-6)/sqrt(x*e + d)

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

[Out]

-2/63/(e*x+d)^(1/2)*(-7*b^5*e^5*x^5-45*a*b^4*e^5*x^4+10*b^5*d*e^4*x^4-126*a^2*b^3*e^5*x^3+72*a*b^4*d*e^4*x^3-1
6*b^5*d^2*e^3*x^3-210*a^3*b^2*e^5*x^2+252*a^2*b^3*d*e^4*x^2-144*a*b^4*d^2*e^3*x^2+32*b^5*d^3*e^2*x^2-315*a^4*b
*e^5*x+840*a^3*b^2*d*e^4*x-1008*a^2*b^3*d^2*e^3*x+576*a*b^4*d^3*e^2*x-128*b^5*d^4*e*x+63*a^5*e^5-630*a^4*b*d*e
^4+1680*a^3*b^2*d^2*e^3-2016*a^2*b^3*d^3*e^2+1152*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 = 1.19, size = 261, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )}}{63 \, \sqrt {e x + d} e^{6}} \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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)/(sqrt(e*x + d)*e^6)

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mupad [B]  time = 1.49, size = 294, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,b^4\,x^5}{9\,e}-\frac {126\,a^5\,e^5-1260\,a^4\,b\,d\,e^4+3360\,a^3\,b^2\,d^2\,e^3-4032\,a^2\,b^3\,d^3\,e^2+2304\,a\,b^4\,d^4\,e-512\,b^5\,d^5}{63\,b\,e^6}+\frac {10\,b^3\,x^4\,\left (9\,a\,e-2\,b\,d\right )}{63\,e^2}+\frac {x\,\left (630\,a^4\,b\,e^5-1680\,a^3\,b^2\,d\,e^4+2016\,a^2\,b^3\,d^2\,e^3-1152\,a\,b^4\,d^3\,e^2+256\,b^5\,d^4\,e\right )}{63\,b\,e^6}+\frac {4\,b^2\,x^3\,\left (63\,a^2\,e^2-36\,a\,b\,d\,e+8\,b^2\,d^2\right )}{63\,e^3}+\frac {4\,b\,x^2\,\left (105\,a^3\,e^3-126\,a^2\,b\,d\,e^2+72\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{63\,e^4}\right )}{x\,\sqrt {d+e\,x}+\frac {a\,\sqrt {d+e\,x}}{b}} \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)^(3/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*b^4*x^5)/(9*e) - (126*a^5*e^5 - 512*b^5*d^5 - 4032*a^2*b^3*d^3*e^2 + 3360
*a^3*b^2*d^2*e^3 + 2304*a*b^4*d^4*e - 1260*a^4*b*d*e^4)/(63*b*e^6) + (10*b^3*x^4*(9*a*e - 2*b*d))/(63*e^2) + (
x*(630*a^4*b*e^5 + 256*b^5*d^4*e - 1152*a*b^4*d^3*e^2 - 1680*a^3*b^2*d*e^4 + 2016*a^2*b^3*d^2*e^3))/(63*b*e^6)
+ (4*b^2*x^3*(63*a^2*e^2 + 8*b^2*d^2 - 36*a*b*d*e))/(63*e^3) + (4*b*x^2*(105*a^3*e^3 - 16*b^3*d^3 + 72*a*b^2*
d^2*e - 126*a^2*b*d*e^2))/(63*e^4)))/(x*(d + e*x)^(1/2) + (a*(d + e*x)^(1/2))/b)

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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)**(3/2),x)

[Out]

Timed out

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