3.16.84 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\) [1584]

Optimal. Leaf size=200 \[ \frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\frac {b^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 (b d-a e)^4 (d+e x)^6} \]

[Out]

1/9*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)/(e*x+d)^9+1/24*b*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)^2/(e*x+d)^8
+1/84*b^2*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)^3/(e*x+d)^7+1/504*b^3*(b*x+a)^5*((b*x+a)^2)^(1/2)/(-a*e+b*d)^
4/(e*x+d)^6

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Rubi [A]
time = 0.05, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {660, 47, 37} \begin {gather*} \frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{84 (d+e x)^7 (b d-a e)^3}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{24 (d+e x)^8 (b d-a e)^2}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{9 (d+e x)^9 (b d-a e)}+\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5}{504 (d+e x)^6 (b d-a e)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(24*(b*d - a*e)^2*(d + e*x)^8) + (b^2*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(84*(b*d - a*e)^3*(
d + e*x)^7) + (b^3*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(504*(b*d - a*e)^4*(d + e*x)^6)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 660

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)^{10}} \, 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)^{10}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\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)^9} \, dx}{3 b^3 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\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)^8} \, dx}{12 b^2 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\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)^7} \, dx}{84 b (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{24 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{84 (b d-a e)^3 (d+e x)^7}+\frac {b^3 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{504 (b d-a e)^4 (d+e x)^6}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 223, normalized size = 1.12 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (56 a^5 e^5+35 a^4 b e^4 (d+9 e x)+20 a^3 b^2 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^2 b^3 e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 a b^4 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+b^5 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{504 e^6 (a+b x) (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^10,x]

[Out]

-1/504*(Sqrt[(a + b*x)^2]*(56*a^5*e^5 + 35*a^4*b*e^4*(d + 9*e*x) + 20*a^3*b^2*e^3*(d^2 + 9*d*e*x + 36*e^2*x^2)
+ 10*a^2*b^3*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*a*b^4*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2
+ 84*d*e^3*x^3 + 126*e^4*x^4) + b^5*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e
^5*x^5)))/(e^6*(a + b*x)*(d + e*x)^9)

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Maple [A]
time = 0.63, size = 288, normalized size = 1.44

method	result	size
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{5} x^{5}}{4 e}-\frac {b^{4} \left (4 a e +b d \right ) x^{4}}{4 e^{2}}-\frac {b^{3} \left (10 a^{2} e^{2}+4 a b d e +b^{2} d^{2}\right ) x^{3}}{6 e^{3}}-\frac {b^{2} \left (20 e^{3} a^{3}+10 a^{2} b d \,e^{2}+4 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{2}}{14 e^{4}}-\frac {b \left (35 e^{4} a^{4}+20 a^{3} b d \,e^{3}+10 a^{2} b^{2} d^{2} e^{2}+4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x}{56 e^{5}}-\frac {56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}}{504 e^{6}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{9}}\)	\(262\)
gosper	\(-\frac {\left (126 b^{5} e^{5} x^{5}+504 a \,b^{4} e^{5} x^{4}+126 b^{5} d \,e^{4} x^{4}+840 a^{2} b^{3} e^{5} x^{3}+336 a \,b^{4} d \,e^{4} x^{3}+84 b^{5} d^{2} e^{3} x^{3}+720 a^{3} b^{2} e^{5} x^{2}+360 a^{2} b^{3} d \,e^{4} x^{2}+144 a \,b^{4} d^{2} e^{3} x^{2}+36 b^{5} d^{3} e^{2} x^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 a^{2} b^{3} d^{2} e^{3} x +36 a \,b^{4} d^{3} e^{2} x +9 b^{5} d^{4} e x +56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\)	\(288\)
default	\(-\frac {\left (126 b^{5} e^{5} x^{5}+504 a \,b^{4} e^{5} x^{4}+126 b^{5} d \,e^{4} x^{4}+840 a^{2} b^{3} e^{5} x^{3}+336 a \,b^{4} d \,e^{4} x^{3}+84 b^{5} d^{2} e^{3} x^{3}+720 a^{3} b^{2} e^{5} x^{2}+360 a^{2} b^{3} d \,e^{4} x^{2}+144 a \,b^{4} d^{2} e^{3} x^{2}+36 b^{5} d^{3} e^{2} x^{2}+315 a^{4} b \,e^{5} x +180 a^{3} b^{2} d \,e^{4} x +90 a^{2} b^{3} d^{2} e^{3} x +36 a \,b^{4} d^{3} e^{2} x +9 b^{5} d^{4} e x +56 a^{5} e^{5}+35 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+4 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 e^{6} \left (e x +d \right )^{9} \left (b x +a \right )^{5}}\)	\(288\)

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)^10,x,method=_RETURNVERBOSE)

[Out]

-1/504/e^6*(126*b^5*e^5*x^5+504*a*b^4*e^5*x^4+126*b^5*d*e^4*x^4+840*a^2*b^3*e^5*x^3+336*a*b^4*d*e^4*x^3+84*b^5
*d^2*e^3*x^3+720*a^3*b^2*e^5*x^2+360*a^2*b^3*d*e^4*x^2+144*a*b^4*d^2*e^3*x^2+36*b^5*d^3*e^2*x^2+315*a^4*b*e^5*
x+180*a^3*b^2*d*e^4*x+90*a^2*b^3*d^2*e^3*x+36*a*b^4*d^3*e^2*x+9*b^5*d^4*e*x+56*a^5*e^5+35*a^4*b*d*e^4+20*a^3*b
^2*d^2*e^3+10*a^2*b^3*d^3*e^2+4*a*b^4*d^4*e+b^5*d^5)*((b*x+a)^2)^(5/2)/(e*x+d)^9/(b*x+a)^5

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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((b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^10,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(b*d-%e*a>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (156) = 312\).
time = 2.54, size = 318, normalized size = 1.59 \begin {gather*} -\frac {b^{5} d^{5} + {\left (126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}\right )} e^{5} + {\left (126 \, b^{5} d x^{4} + 336 \, a b^{4} d x^{3} + 360 \, a^{2} b^{3} d x^{2} + 180 \, a^{3} b^{2} d x + 35 \, a^{4} b d\right )} e^{4} + 2 \, {\left (42 \, b^{5} d^{2} x^{3} + 72 \, a b^{4} d^{2} x^{2} + 45 \, a^{2} b^{3} d^{2} x + 10 \, a^{3} b^{2} d^{2}\right )} e^{3} + 2 \, {\left (18 \, b^{5} d^{3} x^{2} + 18 \, a b^{4} d^{3} x + 5 \, a^{2} b^{3} d^{3}\right )} e^{2} + {\left (9 \, b^{5} d^{4} x + 4 \, a b^{4} d^{4}\right )} e}{504 \, {\left (x^{9} e^{15} + 9 \, d x^{8} e^{14} + 36 \, d^{2} x^{7} e^{13} + 84 \, d^{3} x^{6} e^{12} + 126 \, d^{4} x^{5} e^{11} + 126 \, d^{5} x^{4} e^{10} + 84 \, d^{6} x^{3} e^{9} + 36 \, d^{7} x^{2} e^{8} + 9 \, d^{8} x e^{7} + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/504*(b^5*d^5 + (126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)*e^5
+ (126*b^5*d*x^4 + 336*a*b^4*d*x^3 + 360*a^2*b^3*d*x^2 + 180*a^3*b^2*d*x + 35*a^4*b*d)*e^4 + 2*(42*b^5*d^2*x^
3 + 72*a*b^4*d^2*x^2 + 45*a^2*b^3*d^2*x + 10*a^3*b^2*d^2)*e^3 + 2*(18*b^5*d^3*x^2 + 18*a*b^4*d^3*x + 5*a^2*b^3
*d^3)*e^2 + (9*b^5*d^4*x + 4*a*b^4*d^4)*e)/(x^9*e^15 + 9*d*x^8*e^14 + 36*d^2*x^7*e^13 + 84*d^3*x^6*e^12 + 126*
d^4*x^5*e^11 + 126*d^5*x^4*e^10 + 84*d^6*x^3*e^9 + 36*d^7*x^2*e^8 + 9*d^8*x*e^7 + d^9*e^6)

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Sympy [F(-1)] Timed out
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)**10,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (156) = 312\).
time = 1.65, size = 381, normalized size = 1.90 \begin {gather*} -\frac {{\left (126 \, b^{5} x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, b^{5} d x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 84 \, b^{5} d^{2} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, b^{5} d^{3} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 9 \, b^{5} d^{4} x e \mathrm {sgn}\left (b x + a\right ) + b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) + 504 \, a b^{4} x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 336 \, a b^{4} d x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 144 \, a b^{4} d^{2} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 36 \, a b^{4} d^{3} x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 840 \, a^{2} b^{3} x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 360 \, a^{2} b^{3} d x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 90 \, a^{2} b^{3} d^{2} x e^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, a^{3} b^{2} x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 180 \, a^{3} b^{2} d x e^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 315 \, a^{4} b x e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 56 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{504 \, {\left (x e + d\right )}^{9}} \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)^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5*e^5*sgn(b*x + a) + 126*b^5*d*x^4*e^4*sgn(b*x + a) + 84*b^5*d^2*x^3*e^3*sgn(b*x + a) + 36*b
^5*d^3*x^2*e^2*sgn(b*x + a) + 9*b^5*d^4*x*e*sgn(b*x + a) + b^5*d^5*sgn(b*x + a) + 504*a*b^4*x^4*e^5*sgn(b*x +
a) + 336*a*b^4*d*x^3*e^4*sgn(b*x + a) + 144*a*b^4*d^2*x^2*e^3*sgn(b*x + a) + 36*a*b^4*d^3*x*e^2*sgn(b*x + a) +
4*a*b^4*d^4*e*sgn(b*x + a) + 840*a^2*b^3*x^3*e^5*sgn(b*x + a) + 360*a^2*b^3*d*x^2*e^4*sgn(b*x + a) + 90*a^2*b
^3*d^2*x*e^3*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) + 720*a^3*b^2*x^2*e^5*sgn(b*x + a) + 180*a^3*b^2*d
*x*e^4*sgn(b*x + a) + 20*a^3*b^2*d^2*e^3*sgn(b*x + a) + 315*a^4*b*x*e^5*sgn(b*x + a) + 35*a^4*b*d*e^4*sgn(b*x
+ a) + 56*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^9

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Mupad [B]
time = 0.76, size = 687, normalized size = 3.44 \begin {gather*} \frac {\left (\frac {4\,b^5\,d-5\,a\,b^4\,e}{5\,e^6}+\frac {b^5\,d}{5\,e^6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}-\frac {\left (\frac {5\,a^4\,b\,e^4-10\,a^3\,b^2\,d\,e^3+10\,a^2\,b^3\,d^2\,e^2-5\,a\,b^4\,d^3\,e+b^5\,d^4}{8\,e^6}+\frac {d\,\left (\frac {-10\,a^3\,b^2\,e^4+10\,a^2\,b^3\,d\,e^3-5\,a\,b^4\,d^2\,e^2+b^5\,d^3\,e}{8\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{8\,e^3}-\frac {b^4\,\left (5\,a\,e-b\,d\right )}{8\,e^3}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-5\,a\,b\,d\,e+b^2\,d^2\right )}{8\,e^4}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^8}-\frac {\left (\frac {10\,a^2\,b^3\,e^2-15\,a\,b^4\,d\,e+6\,b^5\,d^2}{6\,e^6}+\frac {d\,\left (\frac {b^5\,d}{6\,e^5}-\frac {b^4\,\left (5\,a\,e-3\,b\,d\right )}{6\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {a^5}{9\,e}-\frac {d\,\left (\frac {5\,a^4\,b}{9\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {5\,a\,b^4}{9\,e}-\frac {b^5\,d}{9\,e^2}\right )}{e}-\frac {10\,a^2\,b^3}{9\,e}\right )}{e}+\frac {10\,a^3\,b^2}{9\,e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^9}+\frac {\left (\frac {-10\,a^3\,b^2\,e^3+20\,a^2\,b^3\,d\,e^2-15\,a\,b^4\,d^2\,e+4\,b^5\,d^3}{7\,e^6}+\frac {d\,\left (\frac {d\,\left (\frac {b^5\,d}{7\,e^4}-\frac {b^4\,\left (5\,a\,e-2\,b\,d\right )}{7\,e^4}\right )}{e}+\frac {b^3\,\left (10\,a^2\,e^2-10\,a\,b\,d\,e+3\,b^2\,d^2\right )}{7\,e^5}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,e^6\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \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)^10,x)

[Out]

(((4*b^5*d - 5*a*b^4*e)/(5*e^6) + (b^5*d)/(5*e^6))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^5) -
(((b^5*d^4 + 5*a^4*b*e^4 - 10*a^3*b^2*d*e^3 + 10*a^2*b^3*d^2*e^2 - 5*a*b^4*d^3*e)/(8*e^6) + (d*((b^5*d^3*e - 1
0*a^3*b^2*e^4 - 5*a*b^4*d^2*e^2 + 10*a^2*b^3*d*e^3)/(8*e^6) + (d*((d*((b^5*d)/(8*e^3) - (b^4*(5*a*e - b*d))/(8
*e^3)))/e + (b^3*(10*a^2*e^2 + b^2*d^2 - 5*a*b*d*e))/(8*e^4)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b
*x)*(d + e*x)^8) - (((6*b^5*d^2 + 10*a^2*b^3*e^2 - 15*a*b^4*d*e)/(6*e^6) + (d*((b^5*d)/(6*e^5) - (b^4*(5*a*e -
3*b*d))/(6*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^6) - ((a^5/(9*e) - (d*((5*a^4*b)/(
9*e) - (d*((d*((d*((5*a*b^4)/(9*e) - (b^5*d)/(9*e^2)))/e - (10*a^2*b^3)/(9*e)))/e + (10*a^3*b^2)/(9*e)))/e))/e
)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) + (((4*b^5*d^3 - 10*a^3*b^2*e^3 + 20*a^2*b^3*d*e^2
- 15*a*b^4*d^2*e)/(7*e^6) + (d*((d*((b^5*d)/(7*e^4) - (b^4*(5*a*e - 2*b*d))/(7*e^4)))/e + (b^3*(10*a^2*e^2 + 3
*b^2*d^2 - 10*a*b*d*e))/(7*e^5)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) - (b^5*(a^2 + b^
2*x^2 + 2*a*b*x)^(1/2))/(4*e^6*(a + b*x)*(d + e*x)^4)

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