Optimal. Leaf size=274 \[ \frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}-\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
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Rubi [A] time = 0.178594, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {664, 612, 621, 206} \[ \frac{3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}-\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 664
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx &=\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac{\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{2 e^2}\\ &=\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}+\frac{\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c d e^2}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac{\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 e^3}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac{\left (3 \left (c d^2-a e^2\right )^5\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^2 d^2 e^3}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac{1}{16} \left (\frac{a}{c d}-\frac{d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}\\ \end{align*}
Mathematica [A] time = 1.24583, size = 384, normalized size = 1.4 \[ \frac{\sqrt{c d} \left (\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left (2 a^3 c^2 d^2 e^5 \left (64 d^2+268 d e x+129 e^2 x^2\right )+2 a^2 c^3 d^3 e^3 \left (87 d^2 e x-35 d^3+489 d e^2 x^2+292 e^3 x^3\right )+5 a^4 c d e^7 (14 d-e x)-15 a^5 e^9+a c^4 d^4 e \left (54 d^2 e^2 x^2-80 d^3 e x+15 d^4+688 d e^3 x^3+464 e^4 x^4\right )+c^5 d^5 x \left (8 d^2 e^2 x^2-10 d^3 e x+15 d^4+176 d e^3 x^3+128 e^4 x^4\right )\right )-15 \left (c d^2-a e^2\right )^{11/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{640 c^{7/2} d^{7/2} e^{7/2} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.049, size = 1123, normalized size = 4.1 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91429, size = 1789, normalized size = 6.53 \begin{align*} \text{result too large to display} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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