3.2005 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=41 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*(b*d - a*e)*(d + e*x)^7)

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Rubi [A]  time = 0.024597, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {767} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*(b*d - a*e)*(d + e*x)^7)

Rule 767

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Sim
p[(f*g*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(b*(p + 1)*(e*f - d*g)), x] /; FreeQ[{a, b, c, d, e, f, g,
m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0]

Rubi steps

\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e) (d+e x)^7}\\ \end{align*}

Mathematica [B]  time = 0.113827, size = 289, normalized size = 7.05 \[ -\frac{\sqrt{(a+b x)^2} \left (a^2 b^4 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+a^3 b^3 e^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^5 b e^5 (d+7 e x)+a^6 e^6+a b^5 e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )\right )}{7 e^7 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a^3*b^3*e^3*
(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 +
35*e^4*x^4) + a*b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b^6*(
d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)))/(7*e^7*(a + b
*x)*(d + e*x)^7)

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Maple [B]  time = 0.007, size = 386, normalized size = 9.4 \begin{align*} -{\frac{7\,{x}^{6}{b}^{6}{e}^{6}+21\,{x}^{5}a{b}^{5}{e}^{6}+21\,{x}^{5}{b}^{6}d{e}^{5}+35\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+35\,{x}^{4}a{b}^{5}d{e}^{5}+35\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+35\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+35\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+35\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+35\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+21\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+21\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+21\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+21\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+21\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+7\,x{a}^{5}b{e}^{6}+7\,x{a}^{4}{b}^{2}d{e}^{5}+7\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+7\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+7\,xa{b}^{5}{d}^{4}{e}^{2}+7\,x{b}^{6}{d}^{5}e+{a}^{6}{e}^{6}+d{e}^{5}{a}^{5}b+{a}^{4}{b}^{2}{d}^{2}{e}^{4}+{a}^{3}{b}^{3}{d}^{3}{e}^{3}+{a}^{2}{b}^{4}{d}^{4}{e}^{2}+a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{7\, \left ( ex+d \right ) ^{7}{e}^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x)

[Out]

-1/7*(7*b^6*e^6*x^6+21*a*b^5*e^6*x^5+21*b^6*d*e^5*x^5+35*a^2*b^4*e^6*x^4+35*a*b^5*d*e^5*x^4+35*b^6*d^2*e^4*x^4
+35*a^3*b^3*e^6*x^3+35*a^2*b^4*d*e^5*x^3+35*a*b^5*d^2*e^4*x^3+35*b^6*d^3*e^3*x^3+21*a^4*b^2*e^6*x^2+21*a^3*b^3
*d*e^5*x^2+21*a^2*b^4*d^2*e^4*x^2+21*a*b^5*d^3*e^3*x^2+21*b^6*d^4*e^2*x^2+7*a^5*b*e^6*x+7*a^4*b^2*d*e^5*x+7*a^
3*b^3*d^2*e^4*x+7*a^2*b^4*d^3*e^3*x+7*a*b^5*d^4*e^2*x+7*b^6*d^5*e*x+a^6*e^6+a^5*b*d*e^5+a^4*b^2*d^2*e^4+a^3*b^
3*d^3*e^3+a^2*b^4*d^4*e^2+a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^7/e^7/(b*x+a)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58878, size = 787, normalized size = 19.2 \begin{align*} -\frac{7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \,{\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \,{\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \,{\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^
5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*
e^3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a
^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d
*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e
^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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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*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.13979, size = 694, normalized size = 16.93 \begin{align*} -\frac{{\left (7 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 21 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 35 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 35 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 7 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 21 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 35 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 35 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 21 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 7 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 7 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 7 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 7 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{7 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/7*(7*b^6*x^6*e^6*sgn(b*x + a) + 21*b^6*d*x^5*e^5*sgn(b*x + a) + 35*b^6*d^2*x^4*e^4*sgn(b*x + a) + 35*b^6*d^
3*x^3*e^3*sgn(b*x + a) + 21*b^6*d^4*x^2*e^2*sgn(b*x + a) + 7*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) +
21*a*b^5*x^5*e^6*sgn(b*x + a) + 35*a*b^5*d*x^4*e^5*sgn(b*x + a) + 35*a*b^5*d^2*x^3*e^4*sgn(b*x + a) + 21*a*b^
5*d^3*x^2*e^3*sgn(b*x + a) + 7*a*b^5*d^4*x*e^2*sgn(b*x + a) + a*b^5*d^5*e*sgn(b*x + a) + 35*a^2*b^4*x^4*e^6*sg
n(b*x + a) + 35*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 21*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 7*a^2*b^4*d^3*x*e^3*sgn
(b*x + a) + a^2*b^4*d^4*e^2*sgn(b*x + a) + 35*a^3*b^3*x^3*e^6*sgn(b*x + a) + 21*a^3*b^3*d*x^2*e^5*sgn(b*x + a)
+ 7*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + a^3*b^3*d^3*e^3*sgn(b*x + a) + 21*a^4*b^2*x^2*e^6*sgn(b*x + a) + 7*a^4*b
^2*d*x*e^5*sgn(b*x + a) + a^4*b^2*d^2*e^4*sgn(b*x + a) + 7*a^5*b*x*e^6*sgn(b*x + a) + a^5*b*d*e^5*sgn(b*x + a)
+ a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^7