Optimal. Leaf size=227 \[ \frac{2 (c+d x)^{3/2} \left (2 \left (3 a^2 b d^2 (45 d e-16 c f)+20 a^3 d^3 f-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+2 a^3 e \sqrt{c+d x}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]
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Rubi [A] time = 0.257237, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {153, 147, 50, 63, 208} \[ \frac{2 (c+d x)^{3/2} \left (2 \left (3 a^2 b d^2 (45 d e-16 c f)+20 a^3 d^3 f-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d x \left (21 a b d^2 e-4 (b c-a d) (2 a d f-2 b c f+3 b d e)\right )\right )}{315 d^4}+2 a^3 e \sqrt{c+d x}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )+\frac{2 (a+b x)^2 (c+d x)^{3/2} (2 a d f-2 b c f+3 b d e)}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 153
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^3 \sqrt{c+d x} (e+f x)}{x} \, dx &=\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{2 \int \frac{(a+b x)^2 \sqrt{c+d x} \left (\frac{9 a d e}{2}+\frac{3}{2} (3 b d e-2 b c f+2 a d f) x\right )}{x} \, dx}{9 d}\\ &=\frac{2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{4 \int \frac{(a+b x) \sqrt{c+d x} \left (\frac{63}{4} a^2 d^2 e+\frac{3}{4} \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{x} \, dx}{63 d^2}\\ &=\frac{2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 e\right ) \int \frac{\sqrt{c+d x}}{x} \, dx\\ &=2 a^3 e \sqrt{c+d x}+\frac{2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\left (a^3 c e\right ) \int \frac{1}{x \sqrt{c+d x}} \, dx\\ &=2 a^3 e \sqrt{c+d x}+\frac{2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}+\frac{\left (2 a^3 c e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=2 a^3 e \sqrt{c+d x}+\frac{2 (3 b d e-2 b c f+2 a d f) (a+b x)^2 (c+d x)^{3/2}}{21 d^2}+\frac{2 f (a+b x)^3 (c+d x)^{3/2}}{9 d}+\frac{2 (c+d x)^{3/2} \left (2 \left (20 a^3 d^3 f+3 a^2 b d^2 (45 d e-16 c f)-9 a b^2 c d (7 d e-4 c f)+4 b^3 c^2 (3 d e-2 c f)\right )+3 b d \left (21 a b d^2 e-4 (b c-a d) (3 b d e-2 b c f+2 a d f)\right ) x\right )}{315 d^4}-2 a^3 \sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.263712, size = 205, normalized size = 0.9 \[ \frac{2 \left (3 d e \left (35 b (c+d x)^{3/2} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )+105 a^3 d^3 \sqrt{c+d x}-105 a^3 \sqrt{c} d^3 \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )-21 b^2 (c+d x)^{5/2} (2 b c-3 a d)+15 b^3 (c+d x)^{7/2}\right )-f (c+d x)^{3/2} \left (135 b^2 (c+d x)^2 (b c-a d)-189 b (c+d x) (b c-a d)^2+105 (b c-a d)^3-35 b^3 (c+d x)^3\right )\right )}{315 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 301, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{{d}^{4}} \left ( 1/9\,f{b}^{3} \left ( dx+c \right ) ^{9/2}+3/7\, \left ( dx+c \right ) ^{7/2}a{b}^{2}df-3/7\, \left ( dx+c \right ) ^{7/2}{b}^{3}cf+1/7\, \left ( dx+c \right ) ^{7/2}{b}^{3}de+3/5\, \left ( dx+c \right ) ^{5/2}{a}^{2}b{d}^{2}f-6/5\, \left ( dx+c \right ) ^{5/2}a{b}^{2}cdf+3/5\, \left ( dx+c \right ) ^{5/2}a{b}^{2}{d}^{2}e+3/5\, \left ( dx+c \right ) ^{5/2}{b}^{3}{c}^{2}f-2/5\, \left ( dx+c \right ) ^{5/2}{b}^{3}cde+1/3\, \left ( dx+c \right ) ^{3/2}{a}^{3}{d}^{3}f- \left ( dx+c \right ) ^{3/2}{a}^{2}bc{d}^{2}f+ \left ( dx+c \right ) ^{3/2}{a}^{2}b{d}^{3}e+ \left ( dx+c \right ) ^{3/2}a{b}^{2}{c}^{2}df- \left ( dx+c \right ) ^{3/2}a{b}^{2}c{d}^{2}e-1/3\, \left ( dx+c \right ) ^{3/2}{b}^{3}{c}^{3}f+1/3\, \left ( dx+c \right ) ^{3/2}{b}^{3}{c}^{2}de+{a}^{3}{d}^{4}e\sqrt{dx+c}-{a}^{3}\sqrt{c}{d}^{4}e{\it Artanh} \left ({\frac{\sqrt{dx+c}}{\sqrt{c}}} \right ) \right ) } \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 = 1.82981, size = 1419, normalized size = 6.25 \begin{align*} \left [\frac{315 \, a^{3} \sqrt{c} d^{4} e \log \left (\frac{d x - 2 \, \sqrt{d x + c} \sqrt{c} + 2 \, c}{x}\right ) + 2 \,{\left (35 \, b^{3} d^{4} f x^{4} + 5 \,{\left (9 \, b^{3} d^{4} e +{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e -{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \,{\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e -{\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f -{\left (3 \,{\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e -{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt{d x + c}}{315 \, d^{4}}, \frac{2 \,{\left (315 \, a^{3} \sqrt{-c} d^{4} e \arctan \left (\frac{\sqrt{d x + c} \sqrt{-c}}{c}\right ) +{\left (35 \, b^{3} d^{4} f x^{4} + 5 \,{\left (9 \, b^{3} d^{4} e +{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} e -{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 3 \,{\left (8 \, b^{3} c^{3} d - 42 \, a b^{2} c^{2} d^{2} + 105 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} e -{\left (16 \, b^{3} c^{4} - 72 \, a b^{2} c^{3} d + 126 \, a^{2} b c^{2} d^{2} - 105 \, a^{3} c d^{3}\right )} f -{\left (3 \,{\left (4 \, b^{3} c^{2} d^{2} - 21 \, a b^{2} c d^{3} - 105 \, a^{2} b d^{4}\right )} e -{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} f\right )} x\right )} \sqrt{d x + c}\right )}}{315 \, d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.773, size = 274, normalized size = 1.21 \begin{align*} \frac{2 a^{3} c e \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + 2 a^{3} e \sqrt{c + d x} + \frac{2 b^{3} f \left (c + d x\right )^{\frac{9}{2}}}{9 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (3 a b^{2} d f - 3 b^{3} c f + b^{3} d e\right )}{7 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (3 a^{2} b d^{2} f - 6 a b^{2} c d f + 3 a b^{2} d^{2} e + 3 b^{3} c^{2} f - 2 b^{3} c d e\right )}{5 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a^{2} b d^{3} e + 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e - b^{3} c^{3} f + b^{3} c^{2} d e\right )}{3 d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.37989, size = 456, normalized size = 2.01 \begin{align*} \frac{2 \, a^{3} c \arctan \left (\frac{\sqrt{d x + c}}{\sqrt{-c}}\right ) e}{\sqrt{-c}} + \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{3} d^{32} f - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} c d^{32} f + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c^{2} d^{32} f - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{3} d^{32} f + 135 \,{\left (d x + c\right )}^{\frac{7}{2}} a b^{2} d^{33} f - 378 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} c d^{33} f + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{33} f + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a^{2} b d^{34} f - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b c d^{34} f + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} d^{35} f + 45 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{3} d^{33} e - 126 \,{\left (d x + c\right )}^{\frac{5}{2}} b^{3} c d^{33} e + 105 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c^{2} d^{33} e + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} a b^{2} d^{34} e - 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} c d^{34} e + 315 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} b d^{35} e + 315 \, \sqrt{d x + c} a^{3} d^{36} e\right )}}{315 \, d^{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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