Optimal. Leaf size=145 \[ \frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (2 c f+d e)+5 b^2 c (2 c f+7 d e)\right )+3 b d x (-4 a d f+4 b c f+7 b d e)\right )}{105 b^3}+2 c^2 e \sqrt {a+b x}-2 \sqrt {a} c^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b} \]
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Rubi [A] time = 0.09, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (2 c f+d e)+5 b^2 c (2 c f+7 d e)\right )+3 b d x (-4 a d f+4 b c f+7 b d e)\right )}{105 b^3}+2 c^2 e \sqrt {a+b x}-2 \sqrt {a} c^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b} \]
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)^2 (e+f x)}{x} \, dx &=\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b}+\frac {2 \int \frac {\sqrt {a+b x} (c+d x) \left (\frac {7 b c e}{2}+\frac {1}{2} (7 b d e+4 b c f-4 a d f) x\right )}{x} \, dx}{7 b}\\ &=\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b}+\frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (d e+2 c f)+5 b^2 c (7 d e+2 c f)\right )+3 b d (7 b d e+4 b c f-4 a d f) x\right )}{105 b^3}+\left (c^2 e\right ) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=2 c^2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b}+\frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (d e+2 c f)+5 b^2 c (7 d e+2 c f)\right )+3 b d (7 b d e+4 b c f-4 a d f) x\right )}{105 b^3}+\left (a c^2 e\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=2 c^2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b}+\frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (d e+2 c f)+5 b^2 c (7 d e+2 c f)\right )+3 b d (7 b d e+4 b c f-4 a d f) x\right )}{105 b^3}+\frac {\left (2 a c^2 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^2 e \sqrt {a+b x}+\frac {2 f (a+b x)^{3/2} (c+d x)^2}{7 b}+\frac {2 (a+b x)^{3/2} \left (2 \left (4 a^2 d^2 f-7 a b d (d e+2 c f)+5 b^2 c (7 d e+2 c f)\right )+3 b d (7 b d e+4 b c f-4 a d f) x\right )}{105 b^3}-2 \sqrt {a} c^2 e \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 146, normalized size = 1.01 \[ \frac {2 \left (7 b e \left (15 b^2 c^2 \sqrt {a+b x}-15 \sqrt {a} b^2 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+5 d (a+b x)^{3/2} (2 b c-a d)+3 d^2 (a+b x)^{5/2}\right )+f (a+b x)^{3/2} \left (42 d (a+b x) (b c-a d)+35 (b c-a d)^2+15 d^2 (a+b x)^2\right )\right )}{105 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 403, normalized size = 2.78 \[ \left [\frac {105 \, \sqrt {a} b^{3} c^{2} e \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (15 \, b^{3} d^{2} f x^{3} + 3 \, {\left (7 \, b^{3} d^{2} e + {\left (14 \, b^{3} c d + a b^{2} d^{2}\right )} f\right )} x^{2} + 7 \, {\left (15 \, b^{3} c^{2} + 10 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e + {\left (35 \, a b^{2} c^{2} - 28 \, a^{2} b c d + 8 \, a^{3} d^{2}\right )} f + {\left (7 \, {\left (10 \, b^{3} c d + a b^{2} d^{2}\right )} e + {\left (35 \, b^{3} c^{2} + 14 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} f\right )} x\right )} \sqrt {b x + a}}{105 \, b^{3}}, \frac {2 \, {\left (105 \, \sqrt {-a} b^{3} c^{2} e \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, b^{3} d^{2} f x^{3} + 3 \, {\left (7 \, b^{3} d^{2} e + {\left (14 \, b^{3} c d + a b^{2} d^{2}\right )} f\right )} x^{2} + 7 \, {\left (15 \, b^{3} c^{2} + 10 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} e + {\left (35 \, a b^{2} c^{2} - 28 \, a^{2} b c d + 8 \, a^{3} d^{2}\right )} f + {\left (7 \, {\left (10 \, b^{3} c d + a b^{2} d^{2}\right )} e + {\left (35 \, b^{3} c^{2} + 14 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} f\right )} x\right )} \sqrt {b x + a}\right )}}{105 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 201, normalized size = 1.39 \[ \frac {2 \, a c^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) e}{\sqrt {-a}} + \frac {2 \, {\left (35 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{20} c^{2} f + 42 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{19} c d f - 70 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{19} c d f + 15 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{18} d^{2} f - 42 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{18} d^{2} f + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{18} d^{2} f + 105 \, \sqrt {b x + a} b^{21} c^{2} e + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{20} c d e + 21 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{19} d^{2} e - 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{19} d^{2} e\right )}}{105 \, b^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 176, normalized size = 1.21 \[ \frac {-2 \sqrt {a}\, b^{3} c^{2} e \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+2 \sqrt {b x +a}\, b^{3} c^{2} e +\frac {2 \left (b x +a \right )^{\frac {3}{2}} a^{2} d^{2} f}{3}-\frac {4 \left (b x +a \right )^{\frac {3}{2}} a b c d f}{3}-\frac {2 \left (b x +a \right )^{\frac {3}{2}} a b \,d^{2} e}{3}+\frac {2 \left (b x +a \right )^{\frac {3}{2}} b^{2} c^{2} f}{3}+\frac {4 \left (b x +a \right )^{\frac {3}{2}} b^{2} c d e}{3}-\frac {4 \left (b x +a \right )^{\frac {5}{2}} a \,d^{2} f}{5}+\frac {4 \left (b x +a \right )^{\frac {5}{2}} b c d f}{5}+\frac {2 \left (b x +a \right )^{\frac {5}{2}} b \,d^{2} e}{5}+\frac {2 \left (b x +a \right )^{\frac {7}{2}} d^{2} f}{7}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 152, normalized size = 1.05 \[ \sqrt {a} c^{2} e \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2 \, {\left (105 \, \sqrt {b x + a} b^{3} c^{2} e + 15 \, {\left (b x + a\right )}^{\frac {7}{2}} d^{2} f + 21 \, {\left (b d^{2} e + 2 \, {\left (b c d - a d^{2}\right )} f\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 35 \, {\left ({\left (2 \, b^{2} c d - a b d^{2}\right )} e + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{105 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 263, normalized size = 1.81 \[ \left (\frac {2\,b\,d^2\,e-6\,a\,d^2\,f+4\,b\,c\,d\,f}{5\,b^3}+\frac {2\,a\,d^2\,f}{5\,b^3}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (a\,\left (a\,\left (\frac {2\,b\,d^2\,e-6\,a\,d^2\,f+4\,b\,c\,d\,f}{b^3}+\frac {2\,a\,d^2\,f}{b^3}\right )-\frac {2\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e\right )}{b^3}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,f-b\,e\right )}{b^3}\right )\,\sqrt {a+b\,x}+\left (\frac {a\,\left (\frac {2\,b\,d^2\,e-6\,a\,d^2\,f+4\,b\,c\,d\,f}{b^3}+\frac {2\,a\,d^2\,f}{b^3}\right )}{3}-\frac {2\,\left (a\,d-b\,c\right )\,\left (b\,c\,f-3\,a\,d\,f+2\,b\,d\,e\right )}{3\,b^3}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {2\,d^2\,f\,{\left (a+b\,x\right )}^{7/2}}{7\,b^3}+\sqrt {a}\,c^2\,e\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.17, size = 167, normalized size = 1.15 \[ \frac {2 a c^{2} e \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 c^{2} e \sqrt {a + b x} + \frac {2 d^{2} f \left (a + b x\right )^{\frac {7}{2}}}{7 b^{3}} + \frac {2 \left (a + b x\right )^{\frac {5}{2}} \left (- 2 a d^{2} f + 2 b c d f + b d^{2} e\right )}{5 b^{3}} + \frac {2 \left (a + b x\right )^{\frac {3}{2}} \left (a^{2} d^{2} f - 2 a b c d f - a b d^{2} e + b^{2} c^{2} f + 2 b^{2} c d e\right )}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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