Optimal. Leaf size=457 \[ \frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-4 b^3 e^3-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e} \]
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Rubi [A] time = 0.577751, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {734, 832, 814, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-4 b^3 e^3-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e} \]
Antiderivative was successfully verified.
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Rule 734
Rule 832
Rule 814
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{\int \sqrt{d+e x} (b d+(2 c d-b e) x) \sqrt{b x+c x^2} \, dx}{3 e}\\ &=-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \int \frac{\left (\frac{1}{2} b d (c d+3 b e)+\left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{4 \int \frac{-\frac{1}{4} b d \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3\right )-\frac{1}{4} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{315 c^2 e^4}-\frac{\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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^4 \sqrt{b x+c x^2}}-\frac{\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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^4 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{\left (\left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (4 d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-15 b c^2 d^2 e+3 b^2 c d e^2-4 b^3 e^3-6 c e \left (c^2 d^2-b c d e+2 b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{-b} \left (16 c^4 d^4-32 b c^3 d^3 e+9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4\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^4 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{8 \sqrt{-b} d (c d-b e) (2 c d-b e) \left (2 c^2 d^2-2 b c d e-b^2 e^2\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^4 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.36044, size = 463, normalized size = 1.01 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (3 b^2 c e^2 (d+e x)-4 b^3 e^3+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (-6 d^2 e x+8 d^3+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (6 b^2 c^2 d^2 e^2+11 b^3 c d e^3-8 b^4 e^4-17 b c^3 d^3 e+8 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (9 b^2 c^2 d^2 e^2+7 b^3 c d e^3-8 b^4 e^4-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.29, size = 1170, normalized size = 2.6 \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{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \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 ({\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}\, dx \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{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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