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

[Out]

-2*d^3*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-c*f+d*e)/(-c*h+d*g)/(d*x+c)^(1/2)-2*b^3*(d*x+c
)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)/(-a*h+b*g)/(b*x+a)^(1/2)+2*b*(a^2*d^2*f*h-a*b*d^2*
(e*h+f*g)+b^2*(2*d^2*e*g+c^2*f*h-c*d*(e*h+f*g)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+
b*e)/(-c*f+d*e)/(-a*h+b*g)/(-c*h+d*g)/(b*x+a)^(1/2)-4*b*d*EllipticF((-a*h+b*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f*g)^
(1/2)/(b*x+a)^(1/2),(-(-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h+b*g))^(1/2))*((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+
a))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*h+b*g)^(1/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2)/(-(-a*f+b*e)*(h*x+g)/(-e*
h+f*g)/(b*x+a))^(1/2)-2*(a^2*d^2*f*h-a*b*d^2*(e*h+f*g)+b^2*(2*d^2*e*g+c^2*f*h-c*d*(e*h+f*g)))*EllipticE((-a*h+
b*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f*g)^(1/2)/(b*x+a)^(1/2),(-(-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h+b*g))^(1/2))*
(-e*h+f*g)^(1/2)*(d*x+c)^(1/2)*(-(-a*f+b*e)*(h*x+g)/(-e*h+f*g)/(b*x+a))^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)/(-c*f+d*
e)/(-c*h+d*g)/(-a*h+b*g)^(1/2)/((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^(1/2)/(h*x+g)^(1/2)

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Rubi [F]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [B]  time = 17.29, size = 7075, normalized size = 9.00 \[ \text {Result too large to show} \]

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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fricas [F]  time = 6.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{b^{2} d^{2} f h x^{6} + a^{2} c^{2} e g + {\left (b^{2} d^{2} f g + {\left (b^{2} d^{2} e + 2 \, {\left (b^{2} c d + a b d^{2}\right )} f\right )} h\right )} x^{5} + {\left ({\left (b^{2} d^{2} e + 2 \, {\left (b^{2} c d + a b d^{2}\right )} f\right )} g + {\left (2 \, {\left (b^{2} c d + a b d^{2}\right )} e + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f\right )} h\right )} x^{4} + {\left ({\left (2 \, {\left (b^{2} c d + a b d^{2}\right )} e + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f\right )} g + {\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e + 2 \, {\left (a b c^{2} + a^{2} c d\right )} f\right )} h\right )} x^{3} + {\left ({\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e + 2 \, {\left (a b c^{2} + a^{2} c d\right )} f\right )} g + {\left (a^{2} c^{2} f + 2 \, {\left (a b c^{2} + a^{2} c d\right )} e\right )} h\right )} x^{2} + {\left (a^{2} c^{2} e h + {\left (a^{2} c^{2} f + 2 \, {\left (a b c^{2} + a^{2} c d\right )} e\right )} g\right )} x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

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)

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maple [B]  time = 0.25, size = 21102, normalized size = 26.85 \[ \text {output too large to display} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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