Optimal. Leaf size=226 \[ -\frac{2 (a+b x)^{3/2} \left (2 \left (-12 a^2 b d^2 (3 c f+d e)+8 a^3 d^3 f+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac{2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt{a+b x}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]
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Rubi [A] time = 0.251778, antiderivative size = 226, 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} \[ -\frac{2 (a+b x)^{3/2} \left (2 \left (-12 a^2 b d^2 (3 c f+d e)+8 a^3 d^3 f+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac{2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt{a+b x}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]
Antiderivative was successfully verified.
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Rule 153
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (c+d x)^3 (e+f x)}{x} \, dx &=\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}+\frac{2 \int \frac{\sqrt{a+b x} (c+d x)^2 \left (\frac{9 b c e}{2}+\frac{3}{2} (3 b d e+2 b c f-2 a d f) x\right )}{x} \, dx}{9 b}\\ &=\frac{2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}+\frac{4 \int \frac{\sqrt{a+b x} (c+d x) \left (\frac{63}{4} b^2 c^2 e+\frac{3}{4} \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{x} \, dx}{63 b^2}\\ &=\frac{2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\left (c^3 e\right ) \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=2 c^3 e \sqrt{a+b x}+\frac{2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\left (a c^3 e\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=2 c^3 e \sqrt{a+b x}+\frac{2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}+\frac{\left (2 a c^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{b}\\ &=2 c^3 e \sqrt{a+b x}+\frac{2 (3 b d e+2 b c f-2 a d f) (a+b x)^{3/2} (c+d x)^2}{21 b^2}+\frac{2 f (a+b x)^{3/2} (c+d x)^3}{9 b}-\frac{2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (d e+3 c f)-5 b^3 c^2 (27 d e+4 c f)+3 a b^2 c d (21 d e+16 c f)\right )-3 b d \left (21 b^2 c d e+4 (b c-a d) (3 b d e+2 b c f-2 a d f)\right ) x\right )}{315 b^4}-2 \sqrt{a} c^3 e \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.273006, size = 204, normalized size = 0.9 \[ \frac{2 \left (3 b e \left (35 d (a+b x)^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+105 b^3 c^3 \sqrt{a+b x}-105 \sqrt{a} b^3 c^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+21 d^2 (a+b x)^{5/2} (3 b c-2 a d)+15 d^3 (a+b x)^{7/2}\right )+f (a+b x)^{3/2} \left (135 d^2 (a+b x)^2 (b c-a d)+189 d (a+b x) (b c-a d)^2+105 (b c-a d)^3+35 d^3 (a+b x)^3\right )\right )}{315 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 301, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{{b}^{4}} \left ( 1/9\,f{d}^{3} \left ( bx+a \right ) ^{9/2}-3/7\, \left ( bx+a \right ) ^{7/2}a{d}^{3}f+3/7\, \left ( bx+a \right ) ^{7/2}bc{d}^{2}f+1/7\, \left ( bx+a \right ) ^{7/2}b{d}^{3}e+3/5\, \left ( bx+a \right ) ^{5/2}{a}^{2}{d}^{3}f-6/5\, \left ( bx+a \right ) ^{5/2}abc{d}^{2}f-2/5\, \left ( bx+a \right ) ^{5/2}ab{d}^{3}e+3/5\, \left ( bx+a \right ) ^{5/2}{b}^{2}{c}^{2}df+3/5\, \left ( bx+a \right ) ^{5/2}{b}^{2}c{d}^{2}e-1/3\, \left ( bx+a \right ) ^{3/2}{a}^{3}{d}^{3}f+ \left ( bx+a \right ) ^{3/2}{a}^{2}bc{d}^{2}f+1/3\, \left ( bx+a \right ) ^{3/2}{a}^{2}b{d}^{3}e- \left ( bx+a \right ) ^{3/2}a{b}^{2}{c}^{2}df- \left ( bx+a \right ) ^{3/2}a{b}^{2}c{d}^{2}e+1/3\, \left ( bx+a \right ) ^{3/2}{b}^{3}{c}^{3}f+ \left ( bx+a \right ) ^{3/2}{b}^{3}{c}^{2}de+{b}^{4}{c}^{3}e\sqrt{bx+a}-\sqrt{a}{b}^{4}{c}^{3}e{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \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 = 1.42747, size = 1419, normalized size = 6.28 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{4} c^{3} e \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (35 \, b^{4} d^{3} f x^{4} + 5 \,{\left (9 \, b^{4} d^{3} e +{\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e +{\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \,{\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e +{\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f +{\left (3 \,{\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e +{\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt{b x + a}}{315 \, b^{4}}, \frac{2 \,{\left (315 \, \sqrt{-a} b^{4} c^{3} e \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (35 \, b^{4} d^{3} f x^{4} + 5 \,{\left (9 \, b^{4} d^{3} e +{\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \,{\left (3 \,{\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e +{\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \,{\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e +{\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f +{\left (3 \,{\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e +{\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt{b x + a}\right )}}{315 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 26.1571, size = 274, normalized size = 1.21 \begin{align*} \frac{2 a c^{3} e \operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + 2 c^{3} e \sqrt{a + b x} + \frac{2 d^{3} f \left (a + b x\right )^{\frac{9}{2}}}{9 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (- 3 a d^{3} f + 3 b c d^{2} f + b d^{3} e\right )}{7 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (3 a^{2} d^{3} f - 6 a b c d^{2} f - 2 a b d^{3} e + 3 b^{2} c^{2} d f + 3 b^{2} c d^{2} e\right )}{5 b^{4}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}} \left (- a^{3} d^{3} f + 3 a^{2} b c d^{2} f + 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 + 3 b^{3} c^{2} d e\right )}{3 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.56003, size = 456, normalized size = 2.02 \begin{align*} \frac{2 \, a c^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) e}{\sqrt{-a}} + \frac{2 \,{\left (105 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{35} c^{3} f + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{34} c^{2} d f - 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{34} c^{2} d f + 135 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{33} c d^{2} f - 378 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{33} c d^{2} f + 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{33} c d^{2} f + 35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{32} d^{3} f - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{32} d^{3} f + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{32} d^{3} f - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{32} d^{3} f + 315 \, \sqrt{b x + a} b^{36} c^{3} e + 315 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{35} c^{2} d e + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{34} c d^{2} e - 315 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{34} c d^{2} e + 45 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{33} d^{3} e - 126 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{33} d^{3} e + 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{33} d^{3} e\right )}}{315 \, b^{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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