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

Optimal. Leaf size=211 \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{3 c^3 d^3 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{8 g (d+e x)^{3/2} (c d f-a e g)}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*(d + e*x)^(3/2)*(f + g*x)^2)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*g*(c*d*f - a*e*g)*
(d + e*x)^(3/2))/(3*c^2*d^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*g*(2*a*e^2*g - c
*d*(e*f + d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^3*d^3*(c*d^2 - a*e^2)*Sqrt[d + e*x])

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Rubi [A]  time = 0.219514, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {866, 788, 648} \[ -\frac{8 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (2 a e^2 g-c d (d g+e f)\right )}{3 c^3 d^3 \sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{8 g (d+e x)^{3/2} (c d f-a e g)}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(d + e*x)^(3/2)*(f + g*x)^2)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*g*(c*d*f - a*e*g)*
(d + e*x)^(3/2))/(3*c^2*d^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (8*g*(2*a*e^2*g - c
*d*(e*f + d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^3*d^3*(c*d^2 - a*e^2)*Sqrt[d + e*x])

Rule 866

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

Rule 788

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

Rule 648

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

Rubi steps

\begin{align*} \int \frac{(d+e x)^{5/2} (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac{(4 g) \int \frac{(d+e x)^{3/2} (f+g x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 g (c d f-a e g) (d+e x)^{3/2}}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{\left (4 g \left (2 a e^2 g-c d (e f+d g)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2 \left (c d^2-a e^2\right )}\\ &=-\frac{2 (d+e x)^{3/2} (f+g x)^2}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac{8 g (c d f-a e g) (d+e x)^{3/2}}{3 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac{8 g \left (2 a e^2 g-c d (e f+d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^3 d^3 \left (c d^2-a e^2\right ) \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0746126, size = 87, normalized size = 0.41 \[ \frac{2 (d+e x)^{3/2} \left (8 a^2 e^2 g^2-4 a c d e g (f-3 g x)-c^2 d^2 \left (f^2+6 f g x-3 g^2 x^2\right )\right )}{3 c^3 d^3 ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(8*a^2*e^2*g^2 - 4*a*c*d*e*g*(f - 3*g*x) - c^2*d^2*(f^2 + 6*f*g*x - 3*g^2*x^2)))/(3*c^3*d^3
*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]  time = 0.05, size = 116, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}+12\,acde{g}^{2}x-6\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-4\,acdefg-{c}^{2}{d}^{2}{f}^{2} \right ) }{3\,{c}^{3}{d}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(c*d*x+a*e)*(3*c^2*d^2*g^2*x^2+12*a*c*d*e*g^2*x-6*c^2*d^2*f*g*x+8*a^2*e^2*g^2-4*a*c*d*e*f*g-c^2*d^2*f^2)*(
e*x+d)^(5/2)/c^3/d^3/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)

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Maxima [A]  time = 1.24443, size = 186, normalized size = 0.88 \begin{align*} -\frac{4 \,{\left (3 \, c d x + 2 \, a e\right )} f g}{3 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt{c d x + a e}} + \frac{2 \,{\left (3 \, c^{2} d^{2} x^{2} + 12 \, a c d e x + 8 \, a^{2} e^{2}\right )} g^{2}}{3 \,{\left (c^{4} d^{4} x + a c^{3} d^{3} e\right )} \sqrt{c d x + a e}} - \frac{2 \, f^{2}}{3 \,{\left (c^{2} d^{2} x + a c d e\right )} \sqrt{c d x + a e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*c*d*x + 2*a*e)*f*g/((c^3*d^3*x + a*c^2*d^2*e)*sqrt(c*d*x + a*e)) + 2/3*(3*c^2*d^2*x^2 + 12*a*c*d*e*x +
8*a^2*e^2)*g^2/((c^4*d^4*x + a*c^3*d^3*e)*sqrt(c*d*x + a*e)) - 2/3*f^2/((c^2*d^2*x + a*c*d*e)*sqrt(c*d*x + a*
e))

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Fricas [A]  time = 1.55553, size = 366, normalized size = 1.73 \begin{align*} \frac{2 \,{\left (3 \, c^{2} d^{2} g^{2} x^{2} - c^{2} d^{2} f^{2} - 4 \, a c d e f g + 8 \, a^{2} e^{2} g^{2} - 6 \,{\left (c^{2} d^{2} f g - 2 \, a c d e g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{3 \,{\left (c^{5} d^{5} e x^{3} + a^{2} c^{3} d^{4} e^{2} +{\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} x^{2} +{\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*c^2*d^2*g^2*x^2 - c^2*d^2*f^2 - 4*a*c*d*e*f*g + 8*a^2*e^2*g^2 - 6*(c^2*d^2*f*g - 2*a*c*d*e*g^2)*x)*sqrt
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x^3 + a^2*c^3*d^4*e^2 + (c^5*d^6 + 2*a*c^4*d^
4*e^2)*x^2 + (2*a*c^4*d^5*e + a^2*c^3*d^3*e^3)*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((e*x+d)**(5/2)*(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x