3.757 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

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Rubi [A]  time = 0.0689705, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {860} \[ \frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

Rule 860

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

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0826987, size = 52, normalized size = 0.83 \[ \frac{2 ((d+e x) (a e+c d x))^{7/2}}{7 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(7/2))/(7*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(7/2))

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Maple [A]  time = 0.049, size = 63, normalized size = 1. \begin{align*} -{\frac{2\,cdx+2\,ae}{7\,aeg-7\,cdf} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( gx+f \right ) ^{-{\frac{7}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x)

[Out]

-2/7*(c*d*x+a*e)/(g*x+f)^(7/2)/(a*e*g-c*d*f)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(e*x+d)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(9/2)), x)

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Fricas [B]  time = 1.52042, size = 625, normalized size = 9.92 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{7 \,{\left (c d^{2} f^{5} - a d e f^{4} g +{\left (c d e f g^{4} - a e^{2} g^{5}\right )} x^{5} +{\left (4 \, c d e f^{2} g^{3} - a d e g^{5} +{\left (c d^{2} - 4 \, a e^{2}\right )} f g^{4}\right )} x^{4} + 2 \,{\left (3 \, c d e f^{3} g^{2} - 2 \, a d e f g^{4} +{\left (2 \, c d^{2} - 3 \, a e^{2}\right )} f^{2} g^{3}\right )} x^{3} + 2 \,{\left (2 \, c d e f^{4} g - 3 \, a d e f^{2} g^{3} +{\left (3 \, c d^{2} - 2 \, a e^{2}\right )} f^{3} g^{2}\right )} x^{2} +{\left (c d e f^{5} - 4 \, a d e f^{3} g^{2} +{\left (4 \, c d^{2} - a e^{2}\right )} f^{4} g\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x, algorithm="fricas")

[Out]

2/7*(c^3*d^3*x^3 + 3*a*c^2*d^2*e*x^2 + 3*a^2*c*d*e^2*x + a^3*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*
sqrt(e*x + d)*sqrt(g*x + f)/(c*d^2*f^5 - a*d*e*f^4*g + (c*d*e*f*g^4 - a*e^2*g^5)*x^5 + (4*c*d*e*f^2*g^3 - a*d*
e*g^5 + (c*d^2 - 4*a*e^2)*f*g^4)*x^4 + 2*(3*c*d*e*f^3*g^2 - 2*a*d*e*f*g^4 + (2*c*d^2 - 3*a*e^2)*f^2*g^3)*x^3 +
2*(2*c*d*e*f^4*g - 3*a*d*e*f^2*g^3 + (3*c*d^2 - 2*a*e^2)*f^3*g^2)*x^2 + (c*d*e*f^5 - 4*a*d*e*f^3*g^2 + (4*c*d
^2 - a*e^2)*f^4*g)*x)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+f)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(9/2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*(g*x + f)^(9/2)), x)