Optimal. Leaf size=611 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right ) \left (5 a d f h \left (3 A d f h^2+C (c h (f g-e h)+d g (e h+2 f g))\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (10 a C d f h (c f h+d e h+d f g)-b \left (15 A d^2 f^2 h^2+C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{4 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a d f h-2 b (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]
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Rubi [A] time = 1.3572, antiderivative size = 608, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1601, 1615, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (e h+2 f g)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{4 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a d f h-2 b (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]
Antiderivative was successfully verified.
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Rule 1601
Rule 1615
Rule 158
Rule 114
Rule 113
Rule 121
Rule 120
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (A+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{\int \frac{-2 b c C e g+5 a A d f h-a C (d e g+c f g+c e h)+(5 A b d f h-3 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h)) x+2 C (a d f h-2 b (d f g+d e h+c f h)) x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{5 d f h}\\ &=\frac{4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \int \frac{\frac{1}{2} d \left (5 a d f h (3 A d f h-C (d e g+c f g+c e h))+2 b C \left (2 d^2 e g (f g+e h)+2 c^2 f h (f g+e h)+c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )\right )+\frac{1}{2} d \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{15 d^3 f^2 h^2}\\ &=\frac{4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{\left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 d^2 f^2 h^3}+\frac{\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{15 d^2 f^2 h^3}\\ &=\frac{4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{\left (\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt{e+f x}}+\frac{\left (\left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}\\ &=\frac{4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \sqrt{-d e+c f} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{\left (\left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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}{15 d^2 f^2 h^3 \sqrt{e+f x} \sqrt{g+h x}}\\ &=\frac{4 C (a d f h-2 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{15 d^2 f^2 h^2}+\frac{2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \sqrt{-d e+c f} \left (15 A b d^2 f^2 h^2-10 a C d f h (d f g+d e h+c f h)+b C \left (8 c^2 f^2 h^2+7 c d f h (f g+e h)+d^2 \left (8 f^2 g^2+7 e f g h+8 e^2 h^2\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{-d e+c f} \left (5 a d f h \left (3 A d f h^2+c C h (f g-e h)+C d g (2 f g+e h)\right )-b \left (15 A d^2 f^2 g h^2+C \left (4 c^2 f h^2 (f g-e h)+c d h \left (3 f^2 g^2+e f g h-4 e^2 h^2\right )+d^2 g \left (8 f^2 g^2+3 e f g h+4 e^2 h^2\right )\right )\right )\right ) \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 )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}\\ \end{align*}
Mathematica [C] time = 9.06159, size = 686, normalized size = 1.12 \[ -\frac{2 \left (-i d h (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right ),\frac{d f g-c f h}{d e h-c f h}\right ) \left (5 a d f h \left (3 A d f^2 h+c C f (e h-f g)+C d e (2 e h+f g)\right )-b \left (15 A d^2 e f^2 h^2+C \left (4 c^2 f^2 h (e h-f g)+c d f \left (3 e^2 h^2+e f g h-4 f^2 g^2\right )+d^2 e \left (8 e^2 h^2+3 e f g h+4 f^2 g^2\right )\right )\right )\right )+d^2 (e+f x) (g+h x) \left (-\sqrt{\frac{d e}{f}-c}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )-i h (c+d x)^{3/2} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right ) \left (-10 a C d f h (c f h+d e h+d f g)+15 A b d^2 f^2 h^2+b C \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )+C d^2 f h (c+d x) (e+f x) (g+h x) \sqrt{\frac{d e}{f}-c} (-5 a d f h+4 b c f h+b d (4 e h+4 f g-3 f h x))\right )}{15 d^4 f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{d e}{f}-c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 5679, normalized size = 9.3 \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{{\left (C x^{2} + A\right )}{\left (b x + a\right )}}{\sqrt{d x + c} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b x^{3} + C a x^{2} + A b x + A a\right )} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}{d f h x^{3} + c e g +{\left (d f g +{\left (d e + c f\right )} h\right )} x^{2} +{\left (c e h +{\left (d e + c f\right )} g\right )} x}, x\right ) \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{{\left (C x^{2} + A\right )}{\left (b x + a\right )}}{\sqrt{d x + c} \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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