3.15.1 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\)

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

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Rubi [A]  time = 0.09, 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)^{3/2}}{3 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {d+e x} (b d-a e)}{e^6 (a+b x)}-\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) \sqrt {d+e x}}+\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^6 (a+b x) (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(2*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)*(d + e*x)^(7/2)) - (2*b*(b*d - a*e)^4*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*(d + e*x)^(5/2)) + (20*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(3*e^6*(a + b*x)*(d + e*x)^(3/2)) - (20*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)*Sqrt[
d + e*x]) - (10*b^4*(b*d - a*e)*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) + (2*b^5*(d + e*x
)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*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)^{9/2}} \, 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)^{9/2}} \, 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)^{9/2}}+\frac {5 b^6 (b d-a e)^4}{e^5 (d+e x)^{7/2}}-\frac {10 b^7 (b d-a e)^3}{e^5 (d+e x)^{5/2}}+\frac {10 b^8 (b d-a e)^2}{e^5 (d+e x)^{3/2}}-\frac {5 b^9 (b d-a e)}{e^5 \sqrt {d+e x}}+\frac {b^{10} \sqrt {d+e x}}{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}}{7 e^6 (a+b x) (d+e x)^{7/2}}-\frac {2 b (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) (d+e x)^{5/2}}+\frac {20 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac {20 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x) \sqrt {d+e x}}-\frac {10 b^4 (b d-a e) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 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 (3 a^5 e^5+3 a^4 b e^4 (2 d+7 e x)+2 a^3 b^2 e^3 \left (8 d^2+28 d e x+35 e^2 x^2\right )+6 a^2 b^3 e^2 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-3 a b^4 e \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(-2*Sqrt[(a + b*x)^2]*(3*a^5*e^5 + 3*a^4*b*e^4*(2*d + 7*e*x) + 2*a^3*b^2*e^3*(8*d^2 + 28*d*e*x + 35*e^2*x^2) +
6*a^2*b^3*e^2*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) - 3*a*b^4*e*(128*d^4 + 448*d^3*e*x + 560*d^2*
e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4) + b^5*(256*d^5 + 896*d^4*e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d
*e^4*x^4 - 7*e^5*x^5)))/(21*e^6*(a + b*x)*(d + e*x)^(7/2))

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IntegrateAlgebraic [A]  time = 26.57, size = 343, normalized size = 1.09 \begin {gather*} \frac {2 \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (-3 a^5 e^5-21 a^4 b e^4 (d+e x)+15 a^4 b d e^4-30 a^3 b^2 d^2 e^3-70 a^3 b^2 e^3 (d+e x)^2+84 a^3 b^2 d e^3 (d+e x)+30 a^2 b^3 d^3 e^2-126 a^2 b^3 d^2 e^2 (d+e x)-210 a^2 b^3 e^2 (d+e x)^3+210 a^2 b^3 d e^2 (d+e x)^2-15 a b^4 d^4 e+84 a b^4 d^3 e (d+e x)-210 a b^4 d^2 e (d+e x)^2+105 a b^4 e (d+e x)^4+420 a b^4 d e (d+e x)^3+3 b^5 d^5-21 b^5 d^4 (d+e x)+70 b^5 d^3 (d+e x)^2-210 b^5 d^2 (d+e x)^3+7 b^5 (d+e x)^5-105 b^5 d (d+e x)^4\right )}{21 e^5 (d+e x)^{7/2} (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)^(9/2),x]

[Out]

(2*Sqrt[(a*e + b*e*x)^2/e^2]*(3*b^5*d^5 - 15*a*b^4*d^4*e + 30*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 15*a^4*b*
d*e^4 - 3*a^5*e^5 - 21*b^5*d^4*(d + e*x) + 84*a*b^4*d^3*e*(d + e*x) - 126*a^2*b^3*d^2*e^2*(d + e*x) + 84*a^3*b
^2*d*e^3*(d + e*x) - 21*a^4*b*e^4*(d + e*x) + 70*b^5*d^3*(d + e*x)^2 - 210*a*b^4*d^2*e*(d + e*x)^2 + 210*a^2*b
^3*d*e^2*(d + e*x)^2 - 70*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 - 105*b^5*d*(d + e*x)^4 + 105*a*b^4*e*(d + e*x)^4 + 7*b^5*(d + e*x)^5))/(21*e^5*(d + e
*x)^(7/2)*(a*e + b*e*x))

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fricas [A]  time = 0.40, size = 304, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\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 )}} \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)^(9/2),x, algorithm="fricas")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 6*a^4*b*d*e^4
- 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 - 70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 7
0*(16*b^5*d^3*e^2 - 24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a*b^4*d^3*e
^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)*sqrt(e*x + d)/(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)

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giac [A]  time = 0.26, size = 456, normalized size = 1.45 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} b^{5} e^{12} \mathrm {sgn}\left (b x + a\right ) - 15 \, \sqrt {x e + d} b^{5} d e^{12} \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {x e + d} a b^{4} e^{13} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-18\right )} - \frac {2 \, {\left (210 \, {\left (x e + d\right )}^{3} b^{5} d^{2} \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (x e + d\right )}^{2} b^{5} d^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )} b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 3 \, b^{5} d^{5} \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 ) + 210 \, {\left (x e + d\right )}^{2} a b^{4} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (x e + d\right )} a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 15 \, 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 ) - 210 \, {\left (x e + d\right )}^{2} a^{2} b^{3} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 126 \, {\left (x e + d\right )} a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 30 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 70 \, {\left (x e + d\right )}^{2} a^{3} b^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 84 \, {\left (x e + d\right )} a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (x e + d\right )} a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right ) - 15 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-6\right )}}{21 \, {\left (x e + d\right )}^{\frac {7}{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)^(9/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^5*e^12*sgn(b*x + a) - 15*sqrt(x*e + d)*b^5*d*e^12*sgn(b*x + a) + 15*sqrt(x*e + d)*a*b^4
*e^13*sgn(b*x + a))*e^(-18) - 2/21*(210*(x*e + d)^3*b^5*d^2*sgn(b*x + a) - 70*(x*e + d)^2*b^5*d^3*sgn(b*x + a)
+ 21*(x*e + d)*b^5*d^4*sgn(b*x + a) - 3*b^5*d^5*sgn(b*x + a) - 420*(x*e + d)^3*a*b^4*d*e*sgn(b*x + a) + 210*(
x*e + d)^2*a*b^4*d^2*e*sgn(b*x + a) - 84*(x*e + d)*a*b^4*d^3*e*sgn(b*x + a) + 15*a*b^4*d^4*e*sgn(b*x + a) + 21
0*(x*e + d)^3*a^2*b^3*e^2*sgn(b*x + a) - 210*(x*e + d)^2*a^2*b^3*d*e^2*sgn(b*x + a) + 126*(x*e + d)*a^2*b^3*d^
2*e^2*sgn(b*x + a) - 30*a^2*b^3*d^3*e^2*sgn(b*x + a) + 70*(x*e + d)^2*a^3*b^2*e^3*sgn(b*x + a) - 84*(x*e + d)*
a^3*b^2*d*e^3*sgn(b*x + a) + 30*a^3*b^2*d^2*e^3*sgn(b*x + a) + 21*(x*e + d)*a^4*b*e^4*sgn(b*x + a) - 15*a^4*b*
d*e^4*sgn(b*x + a) + 3*a^5*e^5*sgn(b*x + a))*e^(-6)/(x*e + d)^(7/2)

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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}-105 a \,b^{4} e^{5} x^{4}+70 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}+560 b^{5} d^{2} e^{3} x^{3}+70 a^{3} b^{2} e^{5} x^{2}+420 a^{2} b^{3} d \,e^{4} x^{2}-1680 a \,b^{4} d^{2} e^{3} x^{2}+1120 b^{5} d^{3} e^{2} x^{2}+21 a^{4} b \,e^{5} x +56 a^{3} b^{2} d \,e^{4} x +336 a^{2} b^{3} d^{2} e^{3} x -1344 a \,b^{4} d^{3} e^{2} x +896 b^{5} d^{4} e x +3 a^{5} e^{5}+6 a^{4} b d \,e^{4}+16 a^{3} b^{2} d^{2} e^{3}+96 a^{2} b^{3} d^{3} e^{2}-384 a \,b^{4} d^{4} e +256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{21 \left (e x +d \right )^{\frac {7}{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)^(9/2),x)

[Out]

-2/21/(e*x+d)^(7/2)*(-7*b^5*e^5*x^5-105*a*b^4*e^5*x^4+70*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
+560*b^5*d^2*e^3*x^3+70*a^3*b^2*e^5*x^2+420*a^2*b^3*d*e^4*x^2-1680*a*b^4*d^2*e^3*x^2+1120*b^5*d^3*e^2*x^2+21*a
^4*b*e^5*x+56*a^3*b^2*d*e^4*x+336*a^2*b^3*d^2*e^3*x-1344*a*b^4*d^3*e^2*x+896*b^5*d^4*e*x+3*a^5*e^5+6*a^4*b*d*e
^4+16*a^3*b^2*d^2*e^3+96*a^2*b^3*d^3*e^2-384*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 = 1.42, size = 293, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )}}{21 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} 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)^(9/2),x, algorithm="maxima")

[Out]

2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 6*a^4*b*d*e^4
- 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^4*e^5)*x^4 - 70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 7
0*(16*b^5*d^3*e^2 - 24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a*b^4*d^3*e
^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^4 + 3*a^4*b*e^5)*x)/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*s
qrt(e*x + d))

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mupad [B]  time = 1.66, size = 386, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {6\,a^5\,e^5+12\,a^4\,b\,d\,e^4+32\,a^3\,b^2\,d^2\,e^3+192\,a^2\,b^3\,d^3\,e^2-768\,a\,b^4\,d^4\,e+512\,b^5\,d^5}{21\,b\,e^9}-\frac {2\,b^4\,x^5}{3\,e^4}-\frac {10\,b^3\,x^4\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^5}+\frac {x\,\left (42\,a^4\,b\,e^5+112\,a^3\,b^2\,d\,e^4+672\,a^2\,b^3\,d^2\,e^3-2688\,a\,b^4\,d^3\,e^2+1792\,b^5\,d^4\,e\right )}{21\,b\,e^9}+\frac {20\,b^2\,x^3\,\left (3\,a^2\,e^2-12\,a\,b\,d\,e+8\,b^2\,d^2\right )}{3\,e^6}+\frac {20\,b\,x^2\,\left (a^3\,e^3+6\,a^2\,b\,d\,e^2-24\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{3\,e^7}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (21\,a\,e^9+63\,b\,d\,e^8\right )\,\sqrt {d+e\,x}}{21\,b\,e^9}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+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)^(9/2),x)

[Out]

-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((6*a^5*e^5 + 512*b^5*d^5 + 192*a^2*b^3*d^3*e^2 + 32*a^3*b^2*d^2*e^3 - 768*a
*b^4*d^4*e + 12*a^4*b*d*e^4)/(21*b*e^9) - (2*b^4*x^5)/(3*e^4) - (10*b^3*x^4*(3*a*e - 2*b*d))/(3*e^5) + (x*(42*
a^4*b*e^5 + 1792*b^5*d^4*e - 2688*a*b^4*d^3*e^2 + 112*a^3*b^2*d*e^4 + 672*a^2*b^3*d^2*e^3))/(21*b*e^9) + (20*b
^2*x^3*(3*a^2*e^2 + 8*b^2*d^2 - 12*a*b*d*e))/(3*e^6) + (20*b*x^2*(a^3*e^3 + 16*b^3*d^3 - 24*a*b^2*d^2*e + 6*a^
2*b*d*e^2))/(3*e^7)))/(x^4*(d + e*x)^(1/2) + (a*d^3*(d + e*x)^(1/2))/(b*e^3) + (x^3*(21*a*e^9 + 63*b*d*e^8)*(d
+ e*x)^(1/2))/(21*b*e^9) + (3*d*x^2*(a*e + b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^2*x*(3*a*e + 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)**(9/2),x)

[Out]

Timed out

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