Optimal. Leaf size=449 \[ \frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 (b c-a d) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 d \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt{h} \sqrt{g+h x} \sqrt{-\frac{f (c+d x)}{d e-c f}}} \]
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Rubi [A] time = 0.671968, antiderivative size = 449, normalized size of antiderivative = 1., 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.
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Rule 179
Rule 121
Rule 120
Rule 169
Rule 538
Rule 537
Rule 114
Rule 113
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*}
Mathematica [C] time = 8.34992, size = 1195, normalized size = 2.66 \[ -\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)}} \text{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-a f) h}{b (e h-f g)};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-a f) h}{b (e h-f g)};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-a f) h}{b (e h-f g)};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.
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Maple [B] time = 0.024, size = 968, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{\frac{3}{2}}}{\left (a + b x\right ) \sqrt{e + f x} \sqrt{g + h x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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