Optimal. Leaf size=226 \[ -\frac {2 (a+b x)^{3/2} \left (2 \left (8 a^3 d^3 f-12 a^2 b d^2 (3 c f+d e)+3 a b^2 c d (16 c f+21 d e)-5 b^3 c^2 (4 c f+27 d e)\right )-3 b d x \left (4 (b c-a d) (-2 a d f+2 b c f+3 b d e)+21 b^2 c d e\right )\right )}{315 b^4}+\frac {2 (a+b x)^{3/2} (c+d x)^2 (-2 a d f+2 b c f+3 b d e)}{21 b^2}+2 c^3 e \sqrt {a+b x}-2 \sqrt {a} c^3 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2} (c+d x)^3}{9 b} \]
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Rubi [A] time = 0.25, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 50
Rule 63
Rule 147
Rule 153
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*}
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Mathematica [A] time = 0.28, size = 204, normalized size = 0.90 \[ \frac {2 \left (3 b e \left (35 d (a+b x)^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+105 b^3 c^3 \sqrt {a+b x}-105 \sqrt {a} b^3 c^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+21 d^2 (a+b x)^{5/2} (3 b c-2 a d)+15 d^3 (a+b x)^{7/2}\right )+f (a+b x)^{3/2} \left (135 d^2 (a+b x)^2 (b c-a d)+189 d (a+b x) (b c-a d)^2+105 (b c-a d)^3+35 d^3 (a+b x)^3\right )\right )}{315 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 641, normalized size = 2.84 \[ \left [\frac {315 \, \sqrt {a} b^{4} c^{3} e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}}{315 \, b^{4}}, \frac {2 \, {\left (315 \, \sqrt {-a} b^{4} c^{3} e \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (35 \, b^{4} d^{3} f x^{4} + 5 \, {\left (9 \, b^{4} d^{3} e + {\left (27 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f\right )} x^{3} + 3 \, {\left (3 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e + {\left (63 \, b^{4} c^{2} d + 9 \, a b^{3} c d^{2} - 2 \, a^{2} b^{2} d^{3}\right )} f\right )} x^{2} + 3 \, {\left (105 \, b^{4} c^{3} + 105 \, a b^{3} c^{2} d - 42 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} e + {\left (105 \, a b^{3} c^{3} - 126 \, a^{2} b^{2} c^{2} d + 72 \, a^{3} b c d^{2} - 16 \, a^{4} d^{3}\right )} f + {\left (3 \, {\left (105 \, b^{4} c^{2} d + 21 \, a b^{3} c d^{2} - 4 \, a^{2} b^{2} d^{3}\right )} e + {\left (105 \, b^{4} c^{3} + 63 \, a b^{3} c^{2} d - 36 \, a^{2} b^{2} c d^{2} + 8 \, a^{3} b d^{3}\right )} f\right )} x\right )} \sqrt {b x + a}\right )}}{315 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 338, normalized size = 1.50 \[ \frac {2 \, a c^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a}} + \frac {2 \, {\left (105 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c^{2} d f - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c^{2} d f + 135 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} c d^{2} f - 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} c d^{2} f + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} c d^{2} f + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{32} d^{3} f - 135 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{32} d^{3} f + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{32} d^{3} f - 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{32} d^{3} f + 315 \, \sqrt {b x + a} b^{36} c^{3} e + 315 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{35} c^{2} d e + 189 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{34} c d^{2} e - 315 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{34} c d^{2} e + 45 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{33} d^{3} e - 126 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{33} d^{3} e + 105 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{33} d^{3} e\right )}}{315 \, b^{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 301, normalized size = 1.33 \[ \frac {-2 \sqrt {a}\, b^{4} c^{3} e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, b^{4} c^{3} e -\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{3} d^{3} f}{3}+2 \left (b x +a \right )^{\frac {3}{2}} a^{2} b c \,d^{2} f +\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{2} b \,d^{3} e}{3}-2 \left (b x +a \right )^{\frac {3}{2}} a \,b^{2} c^{2} d f -2 \left (b x +a \right )^{\frac {3}{2}} a \,b^{2} c \,d^{2} e +\frac {2 \left (b x +a \right )^{\frac {3}{2}} b^{3} c^{3} f}{3}+2 \left (b x +a \right )^{\frac {3}{2}} b^{3} c^{2} d e +\frac {6 \left (b x +a \right )^{\frac {5}{2}} a^{2} d^{3} f}{5}-\frac {12 \left (b x +a \right )^{\frac {5}{2}} a b c \,d^{2} f}{5}-\frac {4 \left (b x +a \right )^{\frac {5}{2}} a b \,d^{3} e}{5}+\frac {6 \left (b x +a \right )^{\frac {5}{2}} b^{2} c^{2} d f}{5}+\frac {6 \left (b x +a \right )^{\frac {5}{2}} b^{2} c \,d^{2} e}{5}-\frac {6 \left (b x +a \right )^{\frac {7}{2}} a \,d^{3} f}{7}+\frac {6 \left (b x +a \right )^{\frac {7}{2}} b c \,d^{2} f}{7}+\frac {2 \left (b x +a \right )^{\frac {7}{2}} b \,d^{3} e}{7}+\frac {2 \left (b x +a \right )^{\frac {9}{2}} d^{3} f}{9}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 238, normalized size = 1.05 \[ \sqrt {a} c^{3} e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (315 \, \sqrt {b x + a} b^{4} c^{3} e + 35 \, {\left (b x + a\right )}^{\frac {9}{2}} d^{3} f + 45 \, {\left (b d^{3} e + 3 \, {\left (b c d^{2} - a d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 63 \, {\left ({\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} e + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 105 \, {\left ({\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{315 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 413, normalized size = 1.83 \[ \left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{7\,b^4}+\frac {2\,a\,d^3\,f}{7\,b^4}\right )\,{\left (a+b\,x\right )}^{7/2}+\left (\frac {a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )}{5}-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{5\,b^4}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (a\,\left (a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{b^4}\right )+\frac {2\,{\left (a\,d-b\,c\right )}^3\,\left (a\,f-b\,e\right )}{b^4}\right )\,\sqrt {a+b\,x}+\left (\frac {a\,\left (a\,\left (\frac {2\,b\,d^3\,e-8\,a\,d^3\,f+6\,b\,c\,d^2\,f}{b^4}+\frac {2\,a\,d^3\,f}{b^4}\right )-\frac {6\,d\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e\right )}{b^4}\right )}{3}+\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (b\,c\,f-4\,a\,d\,f+3\,b\,d\,e\right )}{3\,b^4}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {2\,d^3\,f\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}+\sqrt {a}\,c^3\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.95, size = 274, normalized size = 1.21 \[ \frac {2 a c^{3} e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 c^{3} e \sqrt {a + b x} + \frac {2 d^{3} f \left (a + b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {7}{2}} \left (- 3 a d^{3} f + 3 b c d^{2} f + b d^{3} e\right )}{7 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {5}{2}} \left (3 a^{2} d^{3} f - 6 a b c d^{2} f - 2 a b d^{3} e + 3 b^{2} c^{2} d f + 3 b^{2} c d^{2} e\right )}{5 b^{4}} + \frac {2 \left (a + b x\right )^{\frac {3}{2}} \left (- a^{3} d^{3} f + 3 a^{2} b c d^{2} f + a^{2} b d^{3} e - 3 a b^{2} c^{2} d f - 3 a b^{2} c d^{2} e + b^{3} c^{3} f + 3 b^{3} c^{2} d e\right )}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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