3.1699 \(\int \frac{(a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{9/2}} \, dx\)

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

[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))

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Rubi [A]  time = 0.0942204, antiderivative size = 314, normalized size of antiderivative = 1., 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} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^6 (a+b x)}-\frac{10 b^4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^6 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^6 (a+b x) \sqrt{d+e x}}+\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^6 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^6 (a+b x) (d+e x)^{5/2}}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^6 (a+b x) (d+e x)^{7/2}} \]

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 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]

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])

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*}

Mathematica [A]  time = 0.115255, size = 235, normalized size = 0.75 \[ -\frac{2 \sqrt{(a+b x)^2} \left (6 a^2 b^3 e^2 \left (56 d^2 e x+16 d^3+70 d e^2 x^2+35 e^3 x^3\right )+2 a^3 b^2 e^3 \left (8 d^2+28 d e x+35 e^2 x^2\right )+3 a^4 b e^4 (2 d+7 e x)+3 a^5 e^5-3 a b^4 e \left (560 d^2 e^2 x^2+448 d^3 e x+128 d^4+280 d e^3 x^3+35 e^4 x^4\right )+b^5 \left (1120 d^3 e^2 x^2+560 d^2 e^3 x^3+896 d^4 e x+256 d^5+70 d e^4 x^4-7 e^5 x^5\right )\right )}{21 e^6 (a+b x) (d+e x)^{7/2}} \]

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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Maple [A]  time = 0.157, size = 289, normalized size = 0.9 \begin{align*} -{\frac{-14\,{x}^{5}{b}^{5}{e}^{5}-210\,{x}^{4}a{b}^{4}{e}^{5}+140\,{x}^{4}{b}^{5}d{e}^{4}+420\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-1680\,{x}^{3}a{b}^{4}d{e}^{4}+1120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+140\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+840\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-3360\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}+2240\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+42\,x{a}^{4}b{e}^{5}+112\,x{a}^{3}{b}^{2}d{e}^{4}+672\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-2688\,xa{b}^{4}{d}^{3}{e}^{2}+1792\,x{b}^{5}{d}^{4}e+6\,{a}^{5}{e}^{5}+12\,d{e}^{4}{a}^{4}b+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\, \left ( bx+a \right ) ^{5}{e}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{7}{2}}}} \end{align*}

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 = 1.16313, size = 396, normalized size = 1.26 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.62108, size = 635, normalized size = 2.02 \begin{align*} \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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 1.24065, size = 616, normalized size = 1.96 \begin{align*} \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{align*}

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)