Optimal. Leaf size=379 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]
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Rubi [A] time = 0.502728, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {742, 832, 843, 715, 112, 110, 117, 116} \[ \frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right )}{105 c^3}-\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (24 b^2 e^2-71 b c d e+71 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (6 b^2 e^2-11 b c d e+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{12 e \sqrt{b x+c x^2} (d+e x)^{3/2} (2 c d-b e)}{35 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 832
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{2 \int \frac{(d+e x)^{3/2} \left (\frac{1}{2} d (7 c d-b e)+3 e (2 c d-b e) x\right )}{\sqrt{b x+c x^2}} \, dx}{7 c}\\ &=\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{4 \int \frac{\sqrt{d+e x} \left (\frac{1}{4} d \left (35 c^2 d^2-17 b c d e+6 b^2 e^2\right )+\frac{1}{4} e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) x\right )}{\sqrt{b x+c x^2}} \, dx}{35 c^2}\\ &=\frac{2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{8 \int \frac{\frac{1}{8} d (7 c d-3 b e) \left (15 c^2 d^2-11 b c d e+8 b^2 e^2\right )+e (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^3}\\ &=\frac{2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 b^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{105 c^3}-\frac{\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{105 c^3}\\ &=\frac{2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 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^3 \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 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^3 \sqrt{b x+c x^2}}\\ &=\frac{2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{\left (8 (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 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^3 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{\left (d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 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^3 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 e \left (71 c^2 d^2-71 b c d e+24 b^2 e^2\right ) \sqrt{d+e x} \sqrt{b x+c x^2}}{105 c^3}+\frac{12 e (2 c d-b e) (d+e x)^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{16 \sqrt{-b} (2 c d-b e) \left (11 c^2 d^2-11 b c d e+6 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^{7/2} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}-\frac{2 \sqrt{-b} d (c d-b e) \left (71 c^2 d^2-71 b c d e+24 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^{7/2} \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.22875, size = 388, normalized size = 1.02 \[ \frac{2 \sqrt{x} \left (\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (353 b^2 c^2 d^2 e^2-208 b^3 c d e^3+48 b^4 e^4-298 b c^3 d^3 e+105 c^4 d^4\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )}{b}+e \sqrt{x} (b+c x) (d+e x) \left (24 b^2 e^2-b c e (89 d+18 e x)+c^2 \left (122 d^2+66 d e x+15 e^2 x^2\right )\right )+\frac{8 (b+c x) (d+e x) \left (23 b^2 c d e^2-6 b^3 e^3-33 b c^2 d^2 e+22 c^3 d^3\right )}{c \sqrt{x}}+8 i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (23 b^2 c d e^2-6 b^3 e^3-33 b c^2 d^2 e+22 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )}{105 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.294, size = 918, normalized size = 2.4 \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 (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x}}\,{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 (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b 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 (e x + d\right )}^{\frac{7}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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