Optimal. Leaf size=700 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right ),\frac{h (d e-c f)}{f (d g-c h)}\right ) \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B (c h (f g-e h)+d g (e h+2 f g)))+b^2 \left (-\left (5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))+b^2 \left (-\left (10 A d f h (c f h+d e h+d f g)-B \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (7 a B d f h+b (5 A d f h-4 B (c f h+d e h+d f g)))}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]
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Rubi [A] time = 2.33211, antiderivative size = 699, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1597, 1600, 1615, 158, 114, 113, 121, 120} \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g))+b^2 \left (-\left (5 A d f h (c h (f g-e h)+d g (e h+2 f g))-B \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (c f h+d e h+d f g))+b^2 \left (-\left (10 A d f h (c f h+d e h+d f g)-B \left (8 c^2 f^2 h^2+7 c d f h (e h+f g)+d^2 \left (8 e^2 h^2+7 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (7 a B d f h+5 A b d f h-4 b B (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h} \]
Antiderivative was successfully verified.
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Rule 1597
Rule 1600
Rule 1615
Rule 158
Rule 114
Rule 113
Rule 121
Rule 120
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=\frac{\int \frac{(a+b x) \left (7 a A d f h+7 (A b+a B) d f h x+7 b B d f h x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{7 d f h}\\ &=\frac{2 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{\int \frac{7 d f h \left (5 a^2 A d f h-b B (2 b c e g+a (d e g+c f g+c e h))\right )+7 d f h (5 a (2 A b+a B) d f h-b B (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h))) x+7 b d f h (5 A b d f h+7 a B d f h-4 b 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}{35 d^2 f^2 h^2}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b 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 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 \int \frac{\frac{7}{2} d^2 f h \left (15 a^2 A d^2 f^2 h^2-10 a b B d f h (d e g+c f g+c e h)-b^2 \left (5 A d f h (d e g+c f g+c e h)-B \left (4 d^2 e g (f g+e h)+4 c^2 f h (f g+e h)+2 c d \left (2 f^2 g^2+3 e f g h+2 e^2 h^2\right )\right )\right )\right )+\frac{7}{2} d^2 f h \left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \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 )\right ) x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx}{105 d^4 f^3 h^3}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b 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 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}-\frac{\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \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{\left (15 a^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \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 )\right ) \int \frac{\sqrt{g+h x}}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 d^2 f^2 h^3}\\ &=\frac{2 b (5 A b d f h+7 a B d f h-4 b 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 b B (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}-\frac{\left (\left (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \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^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \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 )\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{2 b (5 A b d f h+7 a B d f h-4 b 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 b B (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^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \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 )\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 (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \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{2 b (5 A b d f h+7 a B d f h-4 b 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 b B (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^2 B d^2 f^2 h^2+10 a b d f h (3 A d f h-2 B (d f g+d e h+c f h))-b^2 \left (10 A d f h (d f g+d e h+c f h)-B \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 )\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 (15 a^2 d^2 f^2 h^2 (B g-A h)+10 a b d f h (3 A d f g h-B c h (f g-e h)-B d g (2 f g+e h))-b^2 \left (5 A d f h (c h (f g-e h)+d g (2 f g+e h))-B \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 = 11.8427, size = 806, normalized size = 1.15 \[ -\frac{2 \left (-\sqrt{\frac{d e}{f}-c} \left (\left (B \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )-10 A d f h (d f g+d e h+c f h)\right ) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right ) (e+f x) (g+h x) d^2+b \sqrt{\frac{d e}{f}-c} f h (c+d x) (e+f x) (g+h x) (-5 A b d f h-10 a B d f h+b B (4 c f h+d (4 f g+4 e h-3 f h x))) d^2-i h \left (-\left (B \left (e \left (4 f^2 g^2+3 e f h g+8 e^2 h^2\right ) d^2+c f \left (-4 f^2 g^2+e f h g+3 e^2 h^2\right ) d+4 c^2 f^2 h (e h-f g)\right )-5 A d f h (c f (e h-f g)+d e (f g+2 e h))\right ) b^2+10 a d f h (-3 A d e f h+B c f (e h-f g)+B d e (f g+2 e h)) b+15 a^2 d^2 f^2 (A f-B e) h^2\right ) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right ),\frac{d f g-c f h}{d e h-c f h}\right ) d-i (d e-c f) h \left (\left (B \left (\left (8 f^2 g^2+7 e f h g+8 e^2 h^2\right ) d^2+7 c f h (f g+e h) d+8 c^2 f^2 h^2\right )-10 A d f h (d f g+d e h+c f h)\right ) b^2-10 a d f h (2 B (d f g+d e h+c f h)-3 A d f h) b+15 a^2 B d^2 f^2 h^2\right ) (c+d x)^{3/2} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )\right )}{15 d^4 \sqrt{\frac{d e}{f}-c} f^3 h^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 8121, normalized size = 11.6 \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 (B x + A\right )}{\left (b x + a\right )}^{2}}{\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 (B b^{2} x^{3} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x\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 (B x + A\right )}{\left (b x + a\right )}^{2}}{\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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