Optimal. Leaf size=597 \[ \frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}+\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (-a^2 b f (2 B d f+7 c C f+13 C d e)+8 a^3 C d f^2+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c \left (-2 A f^2+3 B e f+3 C e^2\right )+A d e f\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]
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Rubi [A] time = 1.35894, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {1614, 150, 158, 114, 113, 121, 120} \[ \frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (-a^2 b f (2 B d f+7 c C f+13 C d e)+8 a^3 C d f^2+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (4 a^2 C f-a b (B f+6 C e)+b^2 (3 B e-2 A f)\right )}{3 b^2 \sqrt{a+b x} (b e-a f)^2}+\frac{2 (d e-c f) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (4 a^2 C d f-a b (B d f+3 C (c f+d e))+b^2 (A d f+3 c C e)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c} (b e-a f)}-\frac{2 (c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]
Antiderivative was successfully verified.
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Rule 1614
Rule 150
Rule 158
Rule 114
Rule 113
Rule 121
Rule 120
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2} \sqrt{e+f x}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{2 \int \frac{\sqrt{c+d x} \left (-\frac{a^2 C (3 d e+c f)+b^2 (3 B c e-2 A c f)-a b (3 c C e+3 B d e+B c f-3 A d f)}{2 b}+\frac{1}{2} \left (a B d f-\frac{4 a^2 C d f}{b}+3 a C (d e+c f)-b (3 c C e+A d f)\right ) x\right )}{(a+b x)^{3/2} \sqrt{e+f x}} \, dx}{3 (b c-a d) (b e-a f)}\\ &=-\frac{2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{4 \int \frac{\frac{4 a^3 C d f (d e+c f)-b^3 c e (3 c C e+3 B d e-A d f)+a b^2 \left (6 c^2 C e f+d^2 e (3 B e-2 A f)+c d \left (9 C e^2+2 B e f+A f^2\right )\right )-a^2 b \left (B d f (d e+c f)+C \left (6 d^2 e^2+11 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac{d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) x}{4 b}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b (b c-a d) (b e-a f)^2}\\ &=-\frac{2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b^2 (b c-a d) f (b e-a f)}-\frac{\left (d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right )\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 b^2 (b c-a d) f (b e-a f)^2}\\ &=-\frac{2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{e+f x}} \, dx}{3 b^2 (b c-a d) f (b e-a f) \sqrt{c+d x}}-\frac{\left (d \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}\right ) \int \frac{\sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx}{3 b^2 (b c-a d) f (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{-b c+a d} f (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left ((d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}} \, dx}{3 b^2 (b c-a d) f (b e-a f) \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 \left (4 a^2 C f+b^2 (3 B e-2 A f)-a b (6 C e+B f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 a^3 C d f^2-a^2 b f (13 C d e+7 c C f+2 B d f)-b^3 \left (3 c C e^2+A d e f+c f (3 B e-2 A f)\right )+a b^2 (3 C e (d e+4 c f)+f (4 B d e+B c f-A d f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{-b c+a d} f (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 (d e-c f) \left (4 a^2 C d f+b^2 (3 c C e+A d f)-a b (B d f+3 C (d e+c f))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^3 \sqrt{d} \sqrt{-b c+a d} f (b e-a f) \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}
Mathematica [C] time = 11.9738, size = 724, normalized size = 1.21 \[ -\frac{2 \left (b^2 f (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left ((a+b x) \left (a^2 b (2 B d f+4 c C f+7 C d e)-5 a^3 C d f-a b^2 (-A d f+B c f+4 B d e+6 c C e)+b^3 (-2 A c f+A d e+3 B c e)\right )+(b c-a d) (b e-a f) \left (a (a C-b B)+A b^2\right )\right )+(a+b x) \left (i b f (a+b x)^{3/2} (b c-a d) (d e-c f) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (-4 a^2 C f+a b (B f+6 C e)+b^2 (2 A f-3 B e)\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left (-a^2 b f (2 B d f+7 c C f+13 C d e)+8 a^3 C d f^2+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right )+i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (-a^2 b f (2 B d f+7 c C f+13 C d e)+8 a^3 C d f^2+a b^2 (f (-A d f+B c f+4 B d e)+3 C e (4 c f+d e))-b^3 \left (c f (3 B e-2 A f)+A d e f+3 c C e^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^4 f (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d) (b e-a f)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.094, size = 13614, normalized size = 22.8 \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{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{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 x^{2} + B x + A\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{3} f x^{4} + a^{3} e +{\left (b^{3} e + 3 \, a b^{2} f\right )} x^{3} + 3 \,{\left (a b^{2} e + a^{2} b f\right )} x^{2} +{\left (3 \, a^{2} b e + a^{3} f\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} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{f x + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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