3.111 \(\int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=786 \[ -\frac{4 b d \sqrt{g+h x} \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e+f x} \sqrt{b g-a h}}{\sqrt{a+b x} \sqrt{f g-e h}}\right ),-\frac{(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right )}{\sqrt{c+d x} (b c-a d)^2 \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 d^2 f h-a b d^2 (e h+f g)+b^2 \left (c^2 f h-c d (e h+f g)+2 d^2 e g\right )\right )}{\sqrt{a+b x} (b c-a d)^2 (b e-a f) (b g-a h) (d e-c f) (d g-c h)}-\frac{2 \sqrt{c+d x} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} \left (a^2 d^2 f h-a b d^2 (e h+f g)+b^2 \left (c^2 f h-c d (e h+f g)+2 d^2 e g\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{g+h x} (b c-a d)^2 (b e-a f) \sqrt{b g-a h} (d e-c f) (d g-c h) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}}-\frac{2 b^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{a+b x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{2 d^3 \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{\sqrt{c+d x} (b c-a d)^2 (d e-c f) (d g-c h)} \]

[Out]

(-2*d^3*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*(d*e - c*f)*(d*g - c*h)*Sqrt[c + d*x]) - (2*
b^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + (2*b*(a
^2*d^2*f*h - a*b*d^2*(f*g + e*h) + b^2*(2*d^2*e*g + c^2*f*h - c*d*(f*g + e*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sq
rt[g + h*x])/((b*c - a*d)^2*(b*e - a*f)*(d*e - c*f)*(b*g - a*h)*(d*g - c*h)*Sqrt[a + b*x]) - (2*Sqrt[f*g - e*h
]*(a^2*d^2*f*h - a*b*d^2*(f*g + e*h) + b^2*(2*d^2*e*g + c^2*f*h - c*d*(f*g + e*h)))*Sqrt[c + d*x]*Sqrt[-(((b*e
- a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]*EllipticE[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]
*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/((b*c - a*d)^2*(b*e - a*f)*(d*e - c
*f)*Sqrt[b*g - a*h]*(d*g - c*h)*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]) - (4*b*d*
Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e +
f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/((b*c - a*d)
^2*Sqrt[b*g - a*h]*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))])

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Rubi [F]  time = 0.010114, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Defer[Int][1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x]

Rubi steps

\begin{align*} \int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\int \frac{1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx\\ \end{align*}

Mathematica [B]  time = 17.4584, size = 7061, normalized size = 8.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x)^(3/2)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.224, size = 21094, normalized size = 26.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{b^{2} d^{2} f h x^{6} + a^{2} c^{2} e g +{\left (b^{2} d^{2} f g +{\left (b^{2} d^{2} e + 2 \,{\left (b^{2} c d + a b d^{2}\right )} f\right )} h\right )} x^{5} +{\left ({\left (b^{2} d^{2} e + 2 \,{\left (b^{2} c d + a b d^{2}\right )} f\right )} g +{\left (2 \,{\left (b^{2} c d + a b d^{2}\right )} e +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f\right )} h\right )} x^{4} +{\left ({\left (2 \,{\left (b^{2} c d + a b d^{2}\right )} e +{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f\right )} g +{\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e + 2 \,{\left (a b c^{2} + a^{2} c d\right )} f\right )} h\right )} x^{3} +{\left ({\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e + 2 \,{\left (a b c^{2} + a^{2} c d\right )} f\right )} g +{\left (a^{2} c^{2} f + 2 \,{\left (a b c^{2} + a^{2} c d\right )} e\right )} h\right )} x^{2} +{\left (a^{2} c^{2} e h +{\left (a^{2} c^{2} f + 2 \,{\left (a b c^{2} + a^{2} c d\right )} e\right )} g\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(b^2*d^2*f*h*x^6 + a^2*c^2*e*g + (b^2*d^2*f*g
+ (b^2*d^2*e + 2*(b^2*c*d + a*b*d^2)*f)*h)*x^5 + ((b^2*d^2*e + 2*(b^2*c*d + a*b*d^2)*f)*g + (2*(b^2*c*d + a*b
*d^2)*e + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f)*h)*x^4 + ((2*(b^2*c*d + a*b*d^2)*e + (b^2*c^2 + 4*a*b*c*d + a^2*d
^2)*f)*g + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e + 2*(a*b*c^2 + a^2*c*d)*f)*h)*x^3 + (((b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*e + 2*(a*b*c^2 + a^2*c*d)*f)*g + (a^2*c^2*f + 2*(a*b*c^2 + a^2*c*d)*e)*h)*x^2 + (a^2*c^2*e*h + (a^2*c^2*
f + 2*(a*b*c^2 + a^2*c*d)*e)*g)*x), 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(1/(b*x+a)**(3/2)/(d*x+c)**(3/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)