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

Optimal. Leaf size=261 \[ -\frac{c d \left (4 a e^2 g-c d (3 d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2} \]

[Out]

-((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) - (
(4*a*e^2*g - c*d*(e*f + 3*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*(c*d*f - a*e*g)^2*Sqrt[d + e
*x]*(f + g*x)) - (c*d*(4*a*e^2*g - c*d*(e*f + 3*d*g))*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(4*g^(3/2)*(c*d*f - a*e*g)^(5/2))

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Rubi [A]  time = 0.355034, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {878, 872, 874, 205} \[ -\frac{c d \left (4 a e^2 g-c d (3 d g+e f)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac{(e f-d g) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt{d+e x} (f+g x)^2 (c d f-a e g)}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt{d+e x} (f+g x) (c d f-a e g)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*f - d*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*g*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) - (
(4*a*e^2*g - c*d*(e*f + 3*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*g*(c*d*f - a*e*g)^2*Sqrt[d + e
*x]*(f + g*x)) - (c*d*(4*a*e^2*g - c*d*(e*f + 3*d*g))*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])/(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])])/(4*g^(3/2)*(c*d*f - a*e*g)^(5/2))

Rule 878

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Simp[(e^2*(e*f - d*g)*(d + e*x)^(m - 2)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p + 1))/(g*(n + 1)*(c*e*f + c*d*g
- b*e*g)), x] - Dist[(e*(b*e*g*(n + 1) + c*e*f*(p + 1) - c*d*g*(2*n + p + 3)))/(g*(n + 1)*(c*e*f + c*d*g - b*
e*g)), Int[(d + e*x)^(m - 1)*(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 - 1, 0] && LtQ[n, -1] && IntegerQ[2*p]

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 874

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[
2*e^2, Subst[Int[1/(c*(e*f + d*g) - b*e*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; Fre
eQ[{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]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{3/2}}{(f+g x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt{d+e x} (f+g x)^2}+\frac{\left (e \left (\frac{1}{2} c d e^2 f+\frac{7}{2} c d^2 e g-2 e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac{\sqrt{d+e x}}{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt{d+e x} (f+g x)^2}-\frac{\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{\left (c d \left (4 a e^2 g-c d (e f+3 d g)\right )\right ) \int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g (c d f-a e g)^2}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt{d+e x} (f+g x)^2}-\frac{\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{\left (c d e^2 \left (4 a e^2 g-c d (e f+3 d g)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{4 g (c d f-a e g)^2}\\ &=-\frac{(e f-d g) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g (c d f-a e g) \sqrt{d+e x} (f+g x)^2}-\frac{\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt{d+e x} (f+g x)}-\frac{c d \left (4 a e^2 g-c d (e f+3 d g)\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d f-a e g} \sqrt{d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.464311, size = 189, normalized size = 0.72 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{c d \left (2 a e^2 g-\frac{1}{2} c d (3 d g+e f)\right ) \left (\frac{c d f-a e g}{c d f+c d g x}+\frac{\sqrt{c d f-a e g} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right )}{\sqrt{g} \sqrt{a e+c d x}}\right )}{(c d f-a e g)^2}+\frac{e f-d g}{(f+g x)^2}\right )}{2 g \sqrt{d+e x} (a e g-c d f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.345, size = 673, normalized size = 2.6 \begin{align*}{\frac{1}{4\, \left ( gx+f \right ) ^{2}g \left ( aeg-cdf \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade} \left ( 4\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}acd{e}^{2}{g}^{3}-3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){x}^{2}{c}^{2}{d}^{3}{g}^{3}-{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ){x}^{2}{c}^{2}{d}^{2}ef{g}^{2}+8\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) xacd{e}^{2}f{g}^{2}-6\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{2}{d}^{3}f{g}^{2}-2\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) x{c}^{2}{d}^{2}e{f}^{2}g+4\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) acd{e}^{2}{f}^{2}g-3\,{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ){c}^{2}{d}^{3}{f}^{2}g-{\it Artanh} \left ({g\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}} \right ){c}^{2}{d}^{2}e{f}^{3}-4\,xa{e}^{2}{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+3\,xc{d}^{2}{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+xcdefg\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-2\,ade{g}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-2\,a{e}^{2}fg\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}+5\,c{d}^{2}fg\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g}-cde{f}^{2}\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{ \left ( aeg-cdf \right ) g}}}{\frac{1}{\sqrt{cdx+ae}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

1/4*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(4*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^2*a*c*d*
e^2*g^3-3*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x^2*c^2*d^3*g^3-arctanh((c*d*x+a*e)^(1/2)*g/((a
*e*g-c*d*f)*g)^(1/2))*x^2*c^2*d^2*e*f*g^2+8*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*a*c*d*e^2*f
*g^2-6*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*x*c^2*d^3*f*g^2-2*arctanh((c*d*x+a*e)^(1/2)*g/((a*
e*g-c*d*f)*g)^(1/2))*x*c^2*d^2*e*f^2*g+4*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*a*c*d*e^2*f^2*g-
3*arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f)*g)^(1/2))*c^2*d^3*f^2*g-arctanh((c*d*x+a*e)^(1/2)*g/((a*e*g-c*d*f
)*g)^(1/2))*c^2*d^2*e*f^3-4*x*a*e^2*g^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*x*c*d^2*g^2*(c*d*x+a*e)^(1
/2)*((a*e*g-c*d*f)*g)^(1/2)+x*c*d*e*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-2*a*d*e*g^2*(c*d*x+a*e)^(1/2
)*((a*e*g-c*d*f)*g)^(1/2)-2*a*e^2*f*g*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+5*c*d^2*f*g*(c*d*x+a*e)^(1/2)*
((a*e*g-c*d*f)*g)^(1/2)-c*d*e*f^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2))/(e*x+d)^(1/2)/((a*e*g-c*d*f)*g)^(
1/2)/(g*x+f)^2/g/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.64706, size = 3411, normalized size = 13.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/8*((c^2*d^3*e*f^3 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f*g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^
3)*x^3 + (2*c^2*d^2*e^2*f^2*g + (7*c^2*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*g^3)*x^2 + (c^
2*d^2*e^2*f^3 + (5*c^2*d^3*e - 4*a*c*d*e^3)*f^2*g + 2*(3*c^2*d^4 - 4*a*c*d^2*e^2)*f*g^2)*x)*sqrt(-c*d*f*g + a*
e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x + 2*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2*(c^2*d^2*e*f
^3*g - 2*a^2*d*e^2*g^4 - (5*c^2*d^3 - a*c*d*e^2)*f^2*g^2 + (7*a*c*d^2*e - 2*a^2*e^3)*f*g^3 - (c^2*d^2*e*f^2*g^
2 + (3*c^2*d^3 - 5*a*c*d*e^2)*f*g^3 - (3*a*c*d^2*e - 4*a^2*e^3)*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^
2)*x)*sqrt(e*x + d))/(c^3*d^4*f^5*g^2 - 3*a*c^2*d^3*e*f^4*g^3 + 3*a^2*c*d^2*e^2*f^3*g^4 - a^3*d*e^3*f^2*g^5 +
(c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^3 + (2*c^3*d^3*e*f^4*g^3 -
a^3*d*e^3*g^7 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^2*g^5 + (3*a^2*c*d^2*
e^2 - 2*a^3*e^4)*f*g^6)*x^2 + (c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 3*a*c^2*d^2*e^2)*f^4*g^3 -
3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^4 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*x), -1/4*((c^2*d^3*e*f^3 + (3*
c^2*d^4 - 4*a*c*d^2*e^2)*f^2*g + (c^2*d^2*e^2*f*g^2 + (3*c^2*d^3*e - 4*a*c*d*e^3)*g^3)*x^3 + (2*c^2*d^2*e^2*f^
2*g + (7*c^2*d^3*e - 8*a*c*d*e^3)*f*g^2 + (3*c^2*d^4 - 4*a*c*d^2*e^2)*g^3)*x^2 + (c^2*d^2*e^2*f^3 + (5*c^2*d^3
*e - 4*a*c*d*e^3)*f^2*g + 2*(3*c^2*d^4 - 4*a*c*d^2*e^2)*f*g^2)*x)*sqrt(c*d*f*g - a*e*g^2)*arctan(sqrt(c*d*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d*e*g*x^2 + a*d*e*g + (c*d^2 + a*e^2)*
g*x)) + (c^2*d^2*e*f^3*g - 2*a^2*d*e^2*g^4 - (5*c^2*d^3 - a*c*d*e^2)*f^2*g^2 + (7*a*c*d^2*e - 2*a^2*e^3)*f*g^3
- (c^2*d^2*e*f^2*g^2 + (3*c^2*d^3 - 5*a*c*d*e^2)*f*g^3 - (3*a*c*d^2*e - 4*a^2*e^3)*g^4)*x)*sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^4*f^5*g^2 - 3*a*c^2*d^3*e*f^4*g^3 + 3*a^2*c*d^2*e^2*f^3*g^4 -
a^3*d*e^3*f^2*g^5 + (c^3*d^3*e*f^3*g^4 - 3*a*c^2*d^2*e^2*f^2*g^5 + 3*a^2*c*d*e^3*f*g^6 - a^3*e^4*g^7)*x^3 + (2
*c^3*d^3*e*f^4*g^3 - a^3*d*e^3*g^7 + (c^3*d^4 - 6*a*c^2*d^2*e^2)*f^3*g^4 - 3*(a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^2
*g^5 + (3*a^2*c*d^2*e^2 - 2*a^3*e^4)*f*g^6)*x^2 + (c^3*d^3*e*f^5*g^2 - 2*a^3*d*e^3*f*g^6 + (2*c^3*d^4 - 3*a*c^
2*d^2*e^2)*f^4*g^3 - 3*(2*a*c^2*d^3*e - a^2*c*d*e^3)*f^3*g^4 + (6*a^2*c*d^2*e^2 - a^3*e^4)*f^2*g^5)*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)**(3/2)/(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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Giac [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)^(3/2)/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

Timed out