Optimal. Leaf size=210 \[ \frac{2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac{2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}-\frac{2 \sqrt{e+f x} (b c-a d)^3}{d^4}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
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Rubi [A] time = 0.177266, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ \frac{2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac{2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}-\frac{2 \sqrt{e+f x} (b c-a d)^3}{d^4}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 \sqrt{e+f x}}{c+d x} \, dx &=\int \left (\frac{b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt{e+f x}}{d^3 f^2}+\frac{(-b c+a d)^3 \sqrt{e+f x}}{d^3 (c+d x)}-\frac{b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{d^2 f^2}+\frac{b^3 (e+f x)^{5/2}}{d f^2}\right ) \, dx\\ &=\frac{2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac{2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac{(b c-a d)^3 \int \frac{\sqrt{e+f x}}{c+d x} \, dx}{d^3}\\ &=-\frac{2 (b c-a d)^3 \sqrt{e+f x}}{d^4}+\frac{2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac{2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac{\left ((b c-a d)^3 (d e-c f)\right ) \int \frac{1}{(c+d x) \sqrt{e+f x}} \, dx}{d^4}\\ &=-\frac{2 (b c-a d)^3 \sqrt{e+f x}}{d^4}+\frac{2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac{2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac{\left (2 (b c-a d)^3 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{d e}{f}+\frac{d x^2}{f}} \, dx,x,\sqrt{e+f x}\right )}{d^4 f}\\ &=-\frac{2 (b c-a d)^3 \sqrt{e+f x}}{d^4}+\frac{2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac{2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.200012, size = 210, normalized size = 1. \[ \frac{2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac{2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}-\frac{2 \sqrt{e+f x} (b c-a d)^3}{d^4}+\frac{2 (b c-a d)^3 \sqrt{d e-c f} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{e+f x}}{\sqrt{d e-c f}}\right )}{d^{9/2}}+\frac{2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 629, normalized size = 3. \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 = 1.49309, size = 1465, normalized size = 6.98 \begin{align*} \left [-\frac{105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt{\frac{d e - c f}{d}} \log \left (\frac{d f x + 2 \, d e - c f - 2 \, \sqrt{f x + e} d \sqrt{\frac{d e - c f}{d}}}{d x + c}\right ) - 2 \,{\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \,{\left (b^{3} d^{3} e f^{2} - 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} -{\left (4 \, b^{3} d^{3} e^{2} f + 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}}{105 \, d^{4} f^{3}}, \frac{2 \,{\left (105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt{-\frac{d e - c f}{d}} \arctan \left (-\frac{\sqrt{f x + e} d \sqrt{-\frac{d e - c f}{d}}}{d e - c f}\right ) +{\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \,{\left (b^{3} d^{3} e f^{2} - 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} -{\left (4 \, b^{3} d^{3} e^{2} f + 7 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt{f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.5202, size = 269, normalized size = 1.28 \begin{align*} \frac{2 \left (\frac{b^{3} \left (e + f x\right )^{\frac{7}{2}}}{7 d f^{2}} + \frac{\left (e + f x\right )^{\frac{5}{2}} \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{5 d^{2} f^{2}} + \frac{\left (e + f x\right )^{\frac{3}{2}} \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{3 d^{3} f^{2}} + \frac{\sqrt{e + f x} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{d^{4}} - \frac{f \left (a d - b c\right )^{3} \left (c f - d e\right ) \operatorname{atan}{\left (\frac{\sqrt{e + f x}}{\sqrt{\frac{c f - d e}{d}}} \right )}}{d^{5} \sqrt{\frac{c f - d e}{d}}}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.12405, size = 589, normalized size = 2.8 \begin{align*} \frac{2 \,{\left (b^{3} c^{4} f - 3 \, a b^{2} c^{3} d f + 3 \, a^{2} b c^{2} d^{2} f - a^{3} c d^{3} f - b^{3} c^{3} d e + 3 \, a b^{2} c^{2} d^{2} e - 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e\right )} \arctan \left (\frac{\sqrt{f x + e} d}{\sqrt{c d f - d^{2} e}}\right )}{\sqrt{c d f - d^{2} e} d^{4}} + \frac{2 \,{\left (15 \,{\left (f x + e\right )}^{\frac{7}{2}} b^{3} d^{6} f^{18} - 21 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} c d^{5} f^{19} + 63 \,{\left (f x + e\right )}^{\frac{5}{2}} a b^{2} d^{6} f^{19} + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c^{2} d^{4} f^{20} - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} c d^{5} f^{20} + 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a^{2} b d^{6} f^{20} - 105 \, \sqrt{f x + e} b^{3} c^{3} d^{3} f^{21} + 315 \, \sqrt{f x + e} a b^{2} c^{2} d^{4} f^{21} - 315 \, \sqrt{f x + e} a^{2} b c d^{5} f^{21} + 105 \, \sqrt{f x + e} a^{3} d^{6} f^{21} - 42 \,{\left (f x + e\right )}^{\frac{5}{2}} b^{3} d^{6} f^{18} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} c d^{5} f^{19} e - 105 \,{\left (f x + e\right )}^{\frac{3}{2}} a b^{2} d^{6} f^{19} e + 35 \,{\left (f x + e\right )}^{\frac{3}{2}} b^{3} d^{6} f^{18} e^{2}\right )}}{105 \, d^{7} f^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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