Optimal. Leaf size=574 \[ \frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-9 b^2 c d e^2-4 b^3 e^3-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (3 b^2 c d e^2+2 b^3 e^3-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]
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Rubi [A] time = 0.782002, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {814, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-3 c e x \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+9 A c e \left (b^2 e^2-11 b c d e+8 c^2 d^2\right )-2 B \left (3 b^2 c d e^2+2 b^3 e^3-42 b c^2 d^2 e+32 c^3 d^3\right )\right )}{315 c^2 e^4}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (9 A c e \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (-9 b^2 c d e^2-4 b^3 e^3-120 b c^2 d^2 e+128 c^3 d^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (\left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right ) \left (9 A c e (2 c d-b e)-B \left (-4 b^2 e^2-7 b c d e+16 c^2 d^2\right )\right )+5 b c d e (2 c d-b e) (-9 A c e-3 b B e+8 B c d)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (-9 A c e-3 b B e+8 B c d-7 B c e x)}{63 c e^2} \]
Antiderivative was successfully verified.
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Rule 814
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx &=-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac{2 \int \frac{\left (-\frac{1}{2} b d (8 B c d-3 b B e-9 A c e)+\frac{1}{2} \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e^2}\\ &=\frac{2 \sqrt{d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^4}-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac{4 \int \frac{-\frac{1}{4} b d \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-84 b c^2 d^2 e+6 b^2 c d e^2+4 b^3 e^3\right )\right )-\frac{1}{4} \left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^4}-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac{\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^5}-\frac{\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{315 c^2 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^4}-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}+\frac{\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{315 c^2 e^5 \sqrt{b x+c x^2}}-\frac{\left (\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{315 c^2 e^5 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^4}-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac{\left (\left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{x} \sqrt{1+\frac{c x}{b}}} \, dx}{315 c^2 e^5 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}}} \, dx}{315 c^2 e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (9 A c e \left (8 c^2 d^2-11 b c d e+b^2 e^2\right )-2 B \left (32 c^3 d^3-42 b c^2 d^2 e+3 b^2 c d e^2+2 b^3 e^3\right )-3 c e \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^4}-\frac{2 \sqrt{d+e x} (8 B c d-3 b B e-9 A c e-7 B c e x) \left (b x+c x^2\right )^{3/2}}{63 c e^2}-\frac{2 \sqrt{-b} \left (5 b c d e (2 c d-b e) (8 B c d-3 b B e-9 A c e)+\left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right ) \left (9 A c e (2 c d-b e)-B \left (16 c^2 d^2-7 b c d e-4 b^2 e^2\right )\right )\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{2 \sqrt{-b} d (c d-b e) \left (9 A c e \left (16 c^2 d^2-16 b c d e-b^2 e^2\right )-B \left (128 c^3 d^3-120 b c^2 d^2 e-9 b^2 c d e^2-4 b^3 e^3\right )\right ) \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^5 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 4.56396, size = 630, normalized size = 1.1 \[ -\frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (B \left (-3 b^2 c e^2 (e x-2 d)+4 b^3 e^3+b c^2 e \left (-84 d^2+61 d e x-50 e^2 x^2\right )+c^3 \left (-48 d^2 e x+64 d^3+40 d e^2 x^2-35 e^3 x^3\right )\right )-9 A c e \left (b^2 e^2+b c e (8 e x-11 d)+c^2 \left (8 d^2-6 d e x+5 e^2 x^2\right )\right )\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (9 A c e \left (-2 b^2 e^2-5 b c d e+8 c^2 d^2\right )+B \left (15 b^2 c d e^2+8 b^3 e^3+36 b c^2 d^2 e-64 c^3 d^3\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (18 A c e \left (2 b^2 c d e^2+b^3 e^3-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (27 b^2 c^2 d^2 e^2+11 b^3 c d e^3+8 b^4 e^4-184 b c^3 d^3 e+128 c^4 d^4\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (18 A c e \left (2 b^2 c d e^2+b^3 e^3-12 b c^2 d^2 e+8 c^3 d^3\right )-B \left (27 b^2 c^2 d^2 e^2+11 b^3 c d e^3+8 b^4 e^4-184 b c^3 d^3 e+128 c^4 d^4\right )\right )\right )\right )}{315 b c^2 e^5 x^2 (b+c x)^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 2112, normalized size = 3.7 \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 (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\sqrt{e x + d}}\,{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 c x^{3} + A b x +{\left (B b + A c\right )} x^{2}\right )} \sqrt{c x^{2} + b x}}{\sqrt{e x + d}}, 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\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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