Optimal. Leaf size=362 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]
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Rubi [A] time = 0.394124, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {742, 814, 843, 715, 112, 110, 117, 116} \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^2 e^2+12 c e x (2 c d-b e)+9 b c d e+3 c^2 d^2\right )}{105 c^2 e}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (8 b^2 e^2-3 b c d e+3 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{5/2} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 e \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 814
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \sqrt{b x+c x^2} \, dx &=\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (\frac{1}{2} d (7 c d-3 b e)+2 e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{7 c}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{4 \int \frac{\frac{1}{4} b d e \left (3 c^2 d^2+9 b c d e-4 b^2 e^2\right )+\frac{1}{4} e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{105 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{105 c^2 e^2 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{105 c^2 e^2 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{\left ((2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\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}{105 c^2 e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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}{105 c^2 e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (3 c^2 d^2+9 b c d e-4 b^2 e^2+12 c e (2 c d-b e) x\right ) \sqrt{b x+c x^2}}{105 c^2 e}+\frac{2 e \sqrt{d+e x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac{2 \sqrt{-b} (2 c d-b e) \left (3 c^2 d^2-3 b c d e+8 b^2 e^2\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 )}{105 c^{5/2} e^2 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \sqrt{-b} d (c d-b e) \left (3 c^2 d^2-3 b c d e+2 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 )}{105 c^{5/2} e^2 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 1.76458, size = 372, normalized size = 1.03 \[ \frac{2 \left (b e x (b+c x) (d+e x) \left (-4 b^2 e^2+3 b c e (3 d+e x)+3 c^2 \left (d^2+8 d e x+5 e^2 x^2\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 c d e^2-8 b^3 e^3-18 b c^2 d^2 e+3 c^3 d^3\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (19 b^2 c d e^2-8 b^3 e^3-9 b c^2 d^2 e+6 c^3 d^3\right )\right )\right )}{105 b c^2 e^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.343, size = 920, normalized size = 2.5 \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 \sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{2}}\,{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 (\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{2}}, 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 \sqrt{x \left (b + c x\right )} \left (d + e x\right )^{\frac{3}{2}}\, 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 \sqrt{c x^{2} + b x}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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