\(\int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [111]

Optimal result

Integrand size = 37, antiderivative size = 786 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 d^3 \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {2 b^3 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (b e-a f) (b g-a h) \sqrt {a+b x}}+\frac {2 b \left (a^2 d^2 f h-a b d^2 (f g+e h)+b^2 \left (2 d^2 e g+c^2 f h-c d (f g+e h)\right )\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (b e-a f) (d e-c f) (b g-a h) (d g-c h) \sqrt {a+b x}}-\frac {2 \sqrt {f g-e h} \left (a^2 d^2 f h-a b d^2 (f g+e h)+b^2 \left (2 d^2 e g+c^2 f h-c d (f g+e h)\right )\right ) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{(b c-a d)^2 (b e-a f) (d e-c f) \sqrt {b g-a h} (d g-c h) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {4 b d \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{(b c-a d)^2 \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]

[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)

Rubi [F]

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

[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*} \text {integral}& = \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 [A] (verified)

Time = 34.40 (sec) , antiderivative size = 670, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {c+d x} \left (-b \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} (e+f x) (g+h x) \left (a^3 d^3 f h-a b^2 d^3 (-e g+f g x+e h x)-a^2 b d^3 (e h+f (g-h x))+b^3 \left (c^3 f h+2 d^3 e g x+c d^2 (e g-f g x-e h x)-c^2 d (f g+e h-f h x)\right )\right )+(c+d x) \left (b^2 \left (a^2 d^2 f h-a b d^2 (f g+e h)+b^2 \left (2 d^2 e g+c^2 f h-c d (f g+e h)\right )\right ) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} (e+f x) (g+h x)+b (f g-e h) (a+b x) \sqrt {-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} \left (\left (a^2 d^2 f h-a b d^2 (f g+e h)+b^2 \left (2 d^2 e g+c^2 f h-c d (f g+e h)\right )\right ) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )-2 b d (d e-c f) (b g-a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )\right )\right )\right )}{b (b c-a d)^2 (b e-a f) (d e-c f) (d g-c h)^2 (a+b x)^{3/2} \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7102\) vs. \(2(724)=1448\).

Time = 3.00 (sec) , antiderivative size = 7103, normalized size of antiderivative = 9.04

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(7103\)
default	\(\text {Expression too large to display}\)	\(22970\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[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)

Sympy [F]

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

[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]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

[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)