3.1981 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=41 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (d+e x)^5 (b d-a e)} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(5*(b*d - a*e)*(d + e*x)^5)

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Rubi [A]  time = 0.0232639, 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} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (d+e x)^5 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/(5*(b*d - a*e)*(d + e*x)^5)

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 )^{3/2}}{(d+e x)^6} \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 (b d-a e) (d+e x)^5}\\ \end{align*}

Mathematica [B]  time = 0.069889, size = 158, normalized size = 3.85 \[ -\frac{\sqrt{(a+b x)^2} \left (a^2 b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+a^3 b e^3 (d+5 e x)+a^4 e^4+a b^3 e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+b^4 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )}{5 e^5 (a+b x) (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^4*e^4 + a^3*b*e^3*(d + 5*e*x) + a^2*b^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + a*b^3*e*(d^3
+ 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + b^4*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))
)/(5*e^5*(a + b*x)*(d + e*x)^5)

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Maple [B]  time = 0.007, size = 197, normalized size = 4.8 \begin{align*} -{\frac{5\,{x}^{4}{b}^{4}{e}^{4}+10\,{x}^{3}a{b}^{3}{e}^{4}+10\,{x}^{3}{b}^{4}d{e}^{3}+10\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+10\,{x}^{2}a{b}^{3}d{e}^{3}+10\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+5\,x{a}^{3}b{e}^{4}+5\,x{a}^{2}{b}^{2}d{e}^{3}+5\,xa{b}^{3}{d}^{2}{e}^{2}+5\,x{b}^{4}{d}^{3}e+{a}^{4}{e}^{4}+d{e}^{3}{a}^{3}b+{a}^{2}{b}^{2}{d}^{2}{e}^{2}+a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{5\, \left ( ex+d \right ) ^{5}{e}^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/(e*x+d)^6,x)

[Out]

-1/5*(5*b^4*e^4*x^4+10*a*b^3*e^4*x^3+10*b^4*d*e^3*x^3+10*a^2*b^2*e^4*x^2+10*a*b^3*d*e^3*x^2+10*b^4*d^2*e^2*x^2
+5*a^3*b*e^4*x+5*a^2*b^2*d*e^3*x+5*a*b^3*d^2*e^2*x+5*b^4*d^3*e*x+a^4*e^4+a^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3
*e+b^4*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^5/e^5/(b*x+a)^3

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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)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62559, size = 428, normalized size = 10.44 \begin{align*} -\frac{5 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + a b^{3} d^{3} e + a^{2} b^{2} d^{2} e^{2} + a^{3} b d e^{3} + a^{4} e^{4} + 10 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 10 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (b^{4} d^{3} e + a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{5 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\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)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

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

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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)**(3/2)/(e*x+d)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.1323, size = 351, normalized size = 8.56 \begin{align*} -\frac{{\left (5 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 5 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 5 \, a^{3} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{5 \,{\left (x e + d\right )}^{5}} \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)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/5*(5*b^4*x^4*e^4*sgn(b*x + a) + 10*b^4*d*x^3*e^3*sgn(b*x + a) + 10*b^4*d^2*x^2*e^2*sgn(b*x + a) + 5*b^4*d^3
*x*e*sgn(b*x + a) + b^4*d^4*sgn(b*x + a) + 10*a*b^3*x^3*e^4*sgn(b*x + a) + 10*a*b^3*d*x^2*e^3*sgn(b*x + a) + 5
*a*b^3*d^2*x*e^2*sgn(b*x + a) + a*b^3*d^3*e*sgn(b*x + a) + 10*a^2*b^2*x^2*e^4*sgn(b*x + a) + 5*a^2*b^2*d*x*e^3
*sgn(b*x + a) + a^2*b^2*d^2*e^2*sgn(b*x + a) + 5*a^3*b*x*e^4*sgn(b*x + a) + a^3*b*d*e^3*sgn(b*x + a) + a^4*e^4
*sgn(b*x + a))*e^(-5)/(x*e + d)^5