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

Optimal. Leaf size=449 \[ \frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 d \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]

[Out]

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

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Rubi [A]  time = 0.67, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {179, 121, 120, 169, 538, 537, 114, 113} \[ \frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 d \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 179

Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])
, x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1
/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx &=\int \left (\frac {d (b c-a d)}{b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {(b c-a d)^2}{b^2 (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {d \sqrt {c+d x}}{b \sqrt {e+f x} \sqrt {g+h x}}\right ) \, dx\\ &=\frac {d \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{b}+\frac {(d (b c-a d)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}+\frac {(b c-a d)^2 \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}\\ &=-\frac {\left (2 (b c-a d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^2}+\frac {\left (d (b c-a d) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{b^2 \sqrt {e+f x}}+\frac {\left (d \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{b \sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}}\\ &=\frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {\left (2 (b c-a d)^2 \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^2 \sqrt {e+f x}}+\frac {\left (d (b c-a d) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{b^2 \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (2 (b c-a d)^2 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {1+\frac {h x^2}{d \left (g-\frac {c h}{d}\right )}}} \, dx,x,\sqrt {c+d x}\right )}{b^2 \sqrt {e+f x} \sqrt {g+h x}}\\ &=\frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}\\ \end {align*}

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Mathematica [C]  time = 7.74, size = 1198, normalized size = 2.67 \[ -\frac {2 \left (b^2 d^2 \sqrt {\frac {f g}{h}-e} h e^3-a b d^2 f \sqrt {\frac {f g}{h}-e} h e^2-b^2 c d f \sqrt {\frac {f g}{h}-e} h e^2-2 b^2 d^2 \sqrt {\frac {f g}{h}-e} h (e+f x) e^2-b^2 d^2 f g \sqrt {\frac {f g}{h}-e} e^2+b^2 d^2 \sqrt {\frac {f g}{h}-e} h (e+f x)^2 e+a b c d f^2 \sqrt {\frac {f g}{h}-e} h e+2 a b d^2 f \sqrt {\frac {f g}{h}-e} h (e+f x) e+b^2 c d f \sqrt {\frac {f g}{h}-e} h (e+f x) e+b^2 d^2 f g \sqrt {\frac {f g}{h}-e} (e+f x) e+a b d^2 f^2 g \sqrt {\frac {f g}{h}-e} e+b^2 c d f^2 g \sqrt {\frac {f g}{h}-e} e-a b d^2 f \sqrt {\frac {f g}{h}-e} h (e+f x)^2-a b c d f^2 \sqrt {\frac {f g}{h}-e} h (e+f x)-a b d^2 f^2 g \sqrt {\frac {f g}{h}-e} (e+f x)+i b d^2 (b e-a f) (f g-e h) \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {f g}{h}-e}}{\sqrt {e+f x}}\right )|\frac {(d e-c f) h}{d (e h-f g)}\right )-i b f \left (a (e h-f g) d^2+b \left (f h c^2-2 d e h c+d^2 e g\right )\right ) \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {f g}{h}-e}}{\sqrt {e+f x}}\right ),\frac {(d e-c f) h}{d (e h-f g)}\right )+i b^2 c^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \Pi \left (-\frac {b e h-a f h}{b f g-b e h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {f g}{h}-e}}{\sqrt {e+f x}}\right )|\frac {(d e-c f) h}{d (e h-f g)}\right )+i a^2 d^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \Pi \left (-\frac {b e h-a f h}{b f g-b e h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {f g}{h}-e}}{\sqrt {e+f x}}\right )|\frac {(d e-c f) h}{d (e h-f g)}\right )-2 i a b c d f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \Pi \left (-\frac {b e h-a f h}{b f g-b e h};i \sinh ^{-1}\left (\frac {\sqrt {\frac {f g}{h}-e}}{\sqrt {e+f x}}\right )|\frac {(d e-c f) h}{d (e h-f g)}\right )-a b c d f^3 g \sqrt {\frac {f g}{h}-e}\right )}{b^2 f^2 (a f-b e) \sqrt {\frac {f g}{h}-e} h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError >> type

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maple [B]  time = 0.03, size = 968, normalized size = 2.16 \[ -\frac {2 \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \left (a c d f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-a c d f h \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-a \,d^{2} e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+a \,d^{2} e h \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+b \,c^{2} f h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-2 b \,c^{2} f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+b \,c^{2} f h \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-b c d e h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+2 b c d e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-b c d e h \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-b c d f g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+b c d f g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+b \,d^{2} e g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-b \,d^{2} e g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )\right )}{\left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right ) b^{2} f h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/h/f/b^2*((d*x+c)/(c*f-d*e)*f)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*
(-(f*x+e)/(c*f-d*e)*d)^(1/2)*(EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c*d*f*h
-EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*d^2*e*h-2*EllipticF(((d*x+c)/(c*f-d*
e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*f*h+2*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h
-d*g)/f*h)^(1/2))*b*c*d*e*h+EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*f*g-E
llipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^2*e*g+EllipticE(((d*x+c)/(c*f-d*e)*f
)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*f*h-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/
f*h)^(1/2))*b*c*d*e*h-EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*f*g+Ellipti
cE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*d^2*e*g-EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/
2),-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*c*d*f*h+EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),
-(c*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*a*d^2*e*h+EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c
*f-d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c^2*f*h-EllipticPi(((d*x+c)/(c*f-d*e)*f)^(1/2),-(c*f-
d*e)/(a*d-b*c)*b/f,((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*b*c*d*e*h)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*
x+c*f*g*x+d*e*g*x+c*e*g)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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