Optimal. Leaf size=875 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 d \sqrt{h} \sqrt{e h-f g} \sqrt{c+d x} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}+\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt{c+d x}}+\frac{4 d \sqrt{f} (d f g+d e h-2 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{f} (2 d f g+d e h-3 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt{e+f x} \sqrt{g+h x}}-\frac{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}+\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}} \]
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Rubi [A] time = 1.33919, antiderivative size = 875, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.343, Rules used = {179, 104, 152, 158, 114, 113, 121, 120, 21, 169, 538, 537} \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 d \sqrt{h} \sqrt{e h-f g} \sqrt{c+d x} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}+\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt{c+d x}}+\frac{4 d \sqrt{f} (d f g+d e h-2 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 \sqrt{f} (2 d f g+d e h-3 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt{e+f x} \sqrt{g+h x}}-\frac{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}+\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 179
Rule 104
Rule 152
Rule 158
Rule 114
Rule 113
Rule 121
Rule 120
Rule 21
Rule 169
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (c+d x)^{5/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\int \left (-\frac{d}{(b c-a d) (c+d x)^{5/2} \sqrt{e+f x} \sqrt{g+h x}}-\frac{b d}{(b c-a d)^2 (c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}}+\frac{b^2}{(b c-a d)^2 (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}\right ) \, dx\\ &=\frac{b^2 \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{(b c-a d)^2}-\frac{(b d) \int \frac{1}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{(b c-a d)^2}-\frac{d \int \frac{1}{(c+d x)^{5/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{b c-a d}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{\left (2 b^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 c-a d)^2}+\frac{(2 b d) \int \frac{-\frac{1}{2} c f h-\frac{1}{2} d f h x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{(b c-a d)^2 (d e-c f) (d g-c h)}+\frac{(2 d) \int \frac{\frac{1}{2} (2 d f g+2 d e h-3 c f h)+\frac{1}{2} d f h x}{(c+d x)^{3/2} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d) (d e-c f) (d g-c h)}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}-\frac{(4 d) \int \frac{-\frac{1}{4} f h \left (d^2 e g-3 c^2 f h+c d (f g+e h)\right )-\frac{1}{2} d f h (d f g+d e h-2 c f h) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2}-\frac{(b d f h) \int \frac{\sqrt{c+d x}}{\sqrt{e+f x} \sqrt{g+h x}} \, dx}{(b c-a d)^2 (d e-c f) (d g-c h)}-\frac{\left (2 b^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 c-a d)^2 \sqrt{e+f x}}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}-\frac{(d f (2 d f g+d e h-3 c f h)) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)}+\frac{\left (2 d^2 f (d f g+d e h-2 c f h)\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2}-\frac{\left (2 b^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 c-a d)^2 \sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (b d f h \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 c-a d)^2 (d e-c f) (d g-c h) \sqrt{\frac{f (c+d x)}{-d e+c f}} \sqrt{g+h x}}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}-\frac{2 b d \sqrt{h} \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 c-a d)^2 (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}-\frac{2 b^2 \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 c-a d)^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (d f (2 d f g+d e h-3 c f h) \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}{3 (b c-a d) (d e-c f)^2 (d g-c h) \sqrt{e+f x}}+\frac{\left (2 d^2 f (d f g+d e h-2 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x}\right ) \int \frac{\sqrt{\frac{d g}{d g-c h}+\frac{d h x}{d g-c h}}}{\sqrt{c+d x} \sqrt{\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}}} \, dx}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}+\frac{4 d \sqrt{f} (d f g+d e h-2 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\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 )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 b d \sqrt{h} \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 c-a d)^2 (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}-\frac{2 b^2 \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 c-a d)^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{\left (d f (2 d f g+d e h-3 c f h) \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}{3 (b c-a d) (d e-c f)^2 (d g-c h) \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{2 d^2 \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac{2 b d^2 \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{4 d^2 (d f g+d e h-2 c f h) \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt{c+d x}}+\frac{4 d \sqrt{f} (d f g+d e h-2 c f h) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\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 )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{2 b d \sqrt{h} \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 c-a d)^2 (d e-c f) (d g-c h) \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}+\frac{2 \sqrt{f} (3 c f h-d (2 f g+e h)) \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 )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h) \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 b^2 \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 c-a d)^3 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}
Mathematica [C] time = 17.4941, size = 12191, normalized size = 13.93 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.257, size = 17330, normalized size = 19.8 \begin{align*} \text{output 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{1}{{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{5}{2}} \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{5}{2}} \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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