Optimal. Leaf size=537 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right ),\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]
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Rubi [A] time = 0.603732, antiderivative size = 537, 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 = {734, 814, 843, 715, 112, 110, 117, 116} \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (3 b^2 e^2-7 c e x (2 c d-b e)-23 b c d e+16 c^2 d^2\right )}{693 c e^3}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-12 c e x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-304 b c^3 d^3 e+128 c^4 d^4\right )}{693 c^2 e^5}+\frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4-256 b c^3 d^3 e+128 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{693 c^{5/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]
Antiderivative was successfully verified.
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Rule 734
Rule 814
Rule 843
Rule 715
Rule 112
Rule 110
Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac{5 \int \frac{(b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 e}\\ &=\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac{10 \int \frac{\left (-\frac{1}{2} b d \left (16 c^2 d^2-23 b c d e+3 b^2 e^2\right )-2 (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c e^3}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac{4 \int \frac{\frac{1}{4} b d \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4\right )+\frac{1}{4} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right ) x}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{693 c^2 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac{\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{b x+c x^2}} \, dx}{693 c^2 e^6}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 b^3 c d e^3+8 b^4 e^4\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx}{693 c^2 e^6}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}+\frac{\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\right ) \sqrt{x} \sqrt{b+c x}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x} \sqrt{d+e x}} \, dx}{693 c^2 e^6 \sqrt{b x+c x^2}}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 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}{693 c^2 e^6 \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 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}{693 c^2 e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{\left (2 d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\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}{693 c^2 e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-304 b c^3 d^3 e+195 b^2 c^2 d^2 e^2-7 b^3 c d e^3-4 b^4 e^4-12 c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right ) \sqrt{b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2-23 b c d e+3 b^2 e^2-7 c e (2 c d-b e) x\right ) \left (b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (b x+c x^2\right )^{5/2}}{11 e}-\frac{2 \sqrt{-b} (2 c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+99 b^2 c^2 d^2 e^2+29 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 )}{693 c^{5/2} e^6 \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}+\frac{4 \sqrt{-b} d (c d-b e) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+5 b^3 c d e^3+2 b^4 e^4\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 )}{693 c^{5/2} e^6 \sqrt{d+e x} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 2.87561, size = 557, normalized size = 1.04 \[ \frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (b^2 c^2 e^2 \left (195 d^2-139 d e x+113 e^2 x^2\right )+b^3 c e^3 (3 e x-7 d)-4 b^4 e^4+b c^3 e \left (224 d^2 e x-304 d^3-185 d e^2 x^2+161 e^3 x^3\right )+c^4 \left (80 d^2 e^2 x^2-96 d^3 e x+128 d^4-70 d e^3 x^3+63 e^4 x^4\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (259 b^2 c^3 d^3 e^2-34 b^3 c^2 d^2 e^3-9 b^4 c d e^4-8 b^5 e^5-336 b c^4 d^4 e+128 c^5 d^5\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right ),\frac{c d}{b e}\right )-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (454 b^2 c^3 d^3 e^2-41 b^3 c^2 d^2 e^3-13 b^4 c d e^4-8 b^5 e^5-640 b c^4 d^4 e+256 c^5 d^5\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-454 b^2 c^3 d^3 e^2+41 b^3 c^2 d^2 e^3+13 b^4 c d e^4+8 b^5 e^5+640 b c^4 d^4 e-256 c^5 d^5\right )\right )\right )}{693 b c^2 e^6 x^3 (b+c x)^3 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.279, size = 1441, normalized size = 2.7 \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{5}{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 (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} 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{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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