3.750 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{13/2}} \, dx\)

Optimal. Leaf size=267 \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)} \]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(11/2)) + (4*c
*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(9/2)) + (16
*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(7/2)
) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*(c*d*f - a*e*g)^4*(d + e*x)^(5/2)*(f + g*
x)^(5/2))

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Rubi [A]  time = 0.323945, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac{32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{1155 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^4}+\frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{231 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^3}+\frac{4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{33 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(13/2)),x]

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(5/2)*(f + g*x)^(11/2)) + (4*c
*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(33*(c*d*f - a*e*g)^2*(d + e*x)^(5/2)*(f + g*x)^(9/2)) + (16
*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(231*(c*d*f - a*e*g)^3*(d + e*x)^(5/2)*(f + g*x)^(7/2)
) + (32*c^3*d^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(1155*(c*d*f - a*e*g)^4*(d + e*x)^(5/2)*(f + g*
x)^(5/2))

Rule 872

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
] - Dist[(c*e*(m - n - 2))/((n + 1)*(c*e*f + c*d*g - b*e*g)), Int[(d + e*x)^m*(f + g*x)^(n + 1)*(a + b*x + c*x
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, 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] && LtQ[n, -1] && IntegerQ[2*p]

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 )^{3/2}}{(d+e x)^{3/2} (f+g x)^{13/2}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac{(6 c d) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{11/2}} \, dx}{11 (c d f-a e g)}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac{\left (8 c^2 d^2\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{9/2}} \, dx}{33 (c d f-a e g)^2}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac{\left (16 c^3 d^3\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{7/2}} \, dx}{231 (c d f-a e g)^3}\\ &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{11 (c d f-a e g) (d+e x)^{5/2} (f+g x)^{11/2}}+\frac{4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{33 (c d f-a e g)^2 (d+e x)^{5/2} (f+g x)^{9/2}}+\frac{16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{231 (c d f-a e g)^3 (d+e x)^{5/2} (f+g x)^{7/2}}+\frac{32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{1155 (c d f-a e g)^4 (d+e x)^{5/2} (f+g x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.186018, size = 152, normalized size = 0.57 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 c d e^2 g^2 (11 f+2 g x)-105 a^3 e^3 g^3-5 a c^2 d^2 e g \left (99 f^2+44 f g x+8 g^2 x^2\right )+c^3 d^3 \left (198 f^2 g x+231 f^3+88 f g^2 x^2+16 g^3 x^3\right )\right )}{1155 (d+e x)^{5/2} (f+g x)^{11/2} (c d f-a e g)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(13/2)),x]

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(-105*a^3*e^3*g^3 + 35*a^2*c*d*e^2*g^2*(11*f + 2*g*x) - 5*a*c^2*d^2*e*g*(99
*f^2 + 44*f*g*x + 8*g^2*x^2) + c^3*d^3*(231*f^3 + 198*f^2*g*x + 88*f*g^2*x^2 + 16*g^3*x^3)))/(1155*(c*d*f - a*
e*g)^4*(d + e*x)^(5/2)*(f + g*x)^(11/2))

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Maple [A]  time = 0.061, size = 260, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}+40\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-88\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-70\,{a}^{2}cd{e}^{2}{g}^{3}x+220\,a{c}^{2}{d}^{2}ef{g}^{2}x-198\,{c}^{3}{d}^{3}{f}^{2}gx+105\,{a}^{3}{e}^{3}{g}^{3}-385\,{a}^{2}cd{e}^{2}f{g}^{2}+495\,a{c}^{2}{d}^{2}e{f}^{2}g-231\,{c}^{3}{d}^{3}{f}^{3} \right ) }{1155\,{g}^{4}{e}^{4}{a}^{4}-4620\,cd{g}^{3}f{e}^{3}{a}^{3}+6930\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-4620\,{c}^{3}{d}^{3}g{f}^{3}ea+1155\,{c}^{4}{d}^{4}{f}^{4}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{11}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x)

[Out]

-2/1155*(c*d*x+a*e)*(-16*c^3*d^3*g^3*x^3+40*a*c^2*d^2*e*g^3*x^2-88*c^3*d^3*f*g^2*x^2-70*a^2*c*d*e^2*g^3*x+220*
a*c^2*d^2*e*f*g^2*x-198*c^3*d^3*f^2*g*x+105*a^3*e^3*g^3-385*a^2*c*d*e^2*f*g^2+495*a*c^2*d^2*e*f^2*g-231*c^3*d^
3*f^3)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)/(g*x+f)^(11/2)/(a^4*e^4*g^4-4*a^3*c*d*e^3*f*g^3+6*a^2*c^2*d^2*e
^2*f^2*g^2-4*a*c^3*d^3*e*f^3*g+c^4*d^4*f^4)/(e*x+d)^(3/2)

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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{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{13}{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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(13/2)), x)

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Fricas [B]  time = 1.79803, size = 2820, normalized size = 10.56 \begin{align*} \text{result too large to display} \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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(16*c^5*d^5*g^3*x^5 + 231*a^2*c^3*d^3*e^2*f^3 - 495*a^3*c^2*d^2*e^3*f^2*g + 385*a^4*c*d*e^4*f*g^2 - 105
*a^5*e^5*g^3 + 8*(11*c^5*d^5*f*g^2 - a*c^4*d^4*e*g^3)*x^4 + 2*(99*c^5*d^5*f^2*g - 22*a*c^4*d^4*e*f*g^2 + 3*a^2
*c^3*d^3*e^2*g^3)*x^3 + (231*c^5*d^5*f^3 - 99*a*c^4*d^4*e*f^2*g + 33*a^2*c^3*d^3*e^2*f*g^2 - 5*a^3*c^2*d^2*e^3
*g^3)*x^2 + 2*(231*a*c^4*d^4*e*f^3 - 396*a^2*c^3*d^3*e^2*f^2*g + 275*a^3*c^2*d^2*e^3*f*g^2 - 70*a^4*c*d*e^4*g^
3)*x)*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^4*d^5*f^10 - 4*a*c^3*d^4*e*f^
9*g + 6*a^2*c^2*d^3*e^2*f^8*g^2 - 4*a^3*c*d^2*e^3*f^7*g^3 + a^4*d*e^4*f^6*g^4 + (c^4*d^4*e*f^4*g^6 - 4*a*c^3*d
^3*e^2*f^3*g^7 + 6*a^2*c^2*d^2*e^3*f^2*g^8 - 4*a^3*c*d*e^4*f*g^9 + a^4*e^5*g^10)*x^7 + (6*c^4*d^4*e*f^5*g^5 +
a^4*d*e^4*g^10 + (c^4*d^5 - 24*a*c^3*d^3*e^2)*f^4*g^6 - 4*(a*c^3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^3*g^7 + 6*(a^2*c
^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^2*g^8 - 2*(2*a^3*c*d^2*e^3 - 3*a^4*e^5)*f*g^9)*x^6 + 3*(5*c^4*d^4*e*f^6*g^4 + 2*
a^4*d*e^4*f*g^9 + 2*(c^4*d^5 - 10*a*c^3*d^3*e^2)*f^5*g^5 - 2*(4*a*c^3*d^4*e - 15*a^2*c^2*d^2*e^3)*f^4*g^6 + 4*
(3*a^2*c^2*d^3*e^2 - 5*a^3*c*d*e^4)*f^3*g^7 - (8*a^3*c*d^2*e^3 - 5*a^4*e^5)*f^2*g^8)*x^5 + 5*(4*c^4*d^4*e*f^7*
g^3 + 3*a^4*d*e^4*f^2*g^8 + (3*c^4*d^5 - 16*a*c^3*d^3*e^2)*f^6*g^4 - 12*(a*c^3*d^4*e - 2*a^2*c^2*d^2*e^3)*f^5*
g^5 + 2*(9*a^2*c^2*d^3*e^2 - 8*a^3*c*d*e^4)*f^4*g^6 - 4*(3*a^3*c*d^2*e^3 - a^4*e^5)*f^3*g^7)*x^4 + 5*(3*c^4*d^
4*e*f^8*g^2 + 4*a^4*d*e^4*f^3*g^7 + 4*(c^4*d^5 - 3*a*c^3*d^3*e^2)*f^7*g^3 - 2*(8*a*c^3*d^4*e - 9*a^2*c^2*d^2*e
^3)*f^6*g^4 + 12*(2*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^5*g^5 - (16*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^4*g^6)*x^3 + 3*(
2*c^4*d^4*e*f^9*g + 5*a^4*d*e^4*f^4*g^6 + (5*c^4*d^5 - 8*a*c^3*d^3*e^2)*f^8*g^2 - 4*(5*a*c^3*d^4*e - 3*a^2*c^2
*d^2*e^3)*f^7*g^3 + 2*(15*a^2*c^2*d^3*e^2 - 4*a^3*c*d*e^4)*f^6*g^4 - 2*(10*a^3*c*d^2*e^3 - a^4*e^5)*f^5*g^5)*x
^2 + (c^4*d^4*e*f^10 + 6*a^4*d*e^4*f^5*g^5 + 2*(3*c^4*d^5 - 2*a*c^3*d^3*e^2)*f^9*g - 6*(4*a*c^3*d^4*e - a^2*c^
2*d^2*e^3)*f^8*g^2 + 4*(9*a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*f^7*g^3 - (24*a^3*c*d^2*e^3 - a^4*e^5)*f^6*g^4)*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)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(13/2),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{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{13}{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)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(13/2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(13/2)), x)