Optimal. Leaf size=146 \[ \frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f+7 a b d (5 d e-2 c f)+b^2 (-c) (7 d e-4 c f)\right )+3 b d x (4 a d f-4 b c f+7 b d e)\right )}{105 d^3}+2 a^2 e \sqrt {c+d x}-2 a^2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d} \]
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Rubi [A] time = 0.10, antiderivative size = 146, 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} \begin {gather*} \frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f+7 a b d (5 d e-2 c f)+b^2 (-c) (7 d e-4 c f)\right )+3 b d x (4 a d f-4 b c f+7 b d e)\right )}{105 d^3}+2 a^2 e \sqrt {c+d x}-2 a^2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 147
Rule 153
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \sqrt {c+d x} (e+f x)}{x} \, dx &=\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d}+\frac {2 \int \frac {(a+b x) \sqrt {c+d x} \left (\frac {7 a d e}{2}+\frac {1}{2} (7 b d e-4 b c f+4 a d f) x\right )}{x} \, dx}{7 d}\\ &=\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f-b^2 c (7 d e-4 c f)+7 a b d (5 d e-2 c f)\right )+3 b d (7 b d e-4 b c f+4 a d f) x\right )}{105 d^3}+\left (a^2 e\right ) \int \frac {\sqrt {c+d x}}{x} \, dx\\ &=2 a^2 e \sqrt {c+d x}+\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f-b^2 c (7 d e-4 c f)+7 a b d (5 d e-2 c f)\right )+3 b d (7 b d e-4 b c f+4 a d f) x\right )}{105 d^3}+\left (a^2 c e\right ) \int \frac {1}{x \sqrt {c+d x}} \, dx\\ &=2 a^2 e \sqrt {c+d x}+\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f-b^2 c (7 d e-4 c f)+7 a b d (5 d e-2 c f)\right )+3 b d (7 b d e-4 b c f+4 a d f) x\right )}{105 d^3}+\frac {\left (2 a^2 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^2 e \sqrt {c+d x}+\frac {2 f (a+b x)^2 (c+d x)^{3/2}}{7 d}+\frac {2 (c+d x)^{3/2} \left (2 \left (10 a^2 d^2 f-b^2 c (7 d e-4 c f)+7 a b d (5 d e-2 c f)\right )+3 b d (7 b d e-4 b c f+4 a d f) x\right )}{105 d^3}-2 a^2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.18, size = 145, normalized size = 0.99 \begin {gather*} \frac {2 \left (7 d e \left (\sqrt {c+d x} \left (15 a^2 d^2+10 a b d (c+d x)+b^2 \left (-2 c^2+c d x+3 d^2 x^2\right )\right )-15 a^2 \sqrt {c} d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )\right )+f (c+d x)^{3/2} \left (-42 b (c+d x) (b c-a d)+35 (b c-a d)^2+15 b^2 (c+d x)^2\right )\right )}{105 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 202, normalized size = 1.38 \begin {gather*} \frac {2 \left (105 a^2 d^3 e \sqrt {c+d x}+35 a^2 d^2 f (c+d x)^{3/2}+70 a b d^2 e (c+d x)^{3/2}+42 a b d f (c+d x)^{5/2}-70 a b c d f (c+d x)^{3/2}+35 b^2 c^2 f (c+d x)^{3/2}+21 b^2 d e (c+d x)^{5/2}-35 b^2 c d e (c+d x)^{3/2}+15 b^2 f (c+d x)^{7/2}-42 b^2 c f (c+d x)^{5/2}\right )}{105 d^3}-2 a^2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.59, size = 405, normalized size = 2.77 \begin {gather*} \left [\frac {105 \, a^{2} \sqrt {c} d^{3} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (15 \, b^{2} d^{3} f x^{3} + 3 \, {\left (7 \, b^{2} d^{3} e + {\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} f\right )} x^{2} - 7 \, {\left (2 \, b^{2} c^{2} d - 10 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} e + {\left (8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} f + {\left (7 \, {\left (b^{2} c d^{2} + 10 \, a b d^{3}\right )} e - {\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} f\right )} x\right )} \sqrt {d x + c}}{105 \, d^{3}}, \frac {2 \, {\left (105 \, a^{2} \sqrt {-c} d^{3} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (15 \, b^{2} d^{3} f x^{3} + 3 \, {\left (7 \, b^{2} d^{3} e + {\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} f\right )} x^{2} - 7 \, {\left (2 \, b^{2} c^{2} d - 10 \, a b c d^{2} - 15 \, a^{2} d^{3}\right )} e + {\left (8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} f + {\left (7 \, {\left (b^{2} c d^{2} + 10 \, a b d^{3}\right )} e - {\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} f\right )} x\right )} \sqrt {d x + c}\right )}}{105 \, d^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 201, normalized size = 1.38 \begin {gather*} \frac {2 \, a^{2} c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{\sqrt {-c}} + \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} d^{18} f - 42 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} c d^{18} f + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c^{2} d^{18} f + 42 \, {\left (d x + c\right )}^{\frac {5}{2}} a b d^{19} f - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a b c d^{19} f + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} d^{20} f + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{19} e - 35 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{19} e + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a b d^{20} e + 105 \, \sqrt {d x + c} a^{2} d^{21} e\right )}}{105 \, d^{21}} \end {gather*}
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 \begin {gather*} \frac {-2 a^{2} \sqrt {c}\, d^{3} e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+2 \sqrt {d x +c}\, a^{2} d^{3} e +\frac {2 \left (d x +c \right )^{\frac {3}{2}} a^{2} d^{2} f}{3}-\frac {4 \left (d x +c \right )^{\frac {3}{2}} a b c d f}{3}+\frac {4 \left (d x +c \right )^{\frac {3}{2}} a b \,d^{2} e}{3}+\frac {2 \left (d x +c \right )^{\frac {3}{2}} b^{2} c^{2} f}{3}-\frac {2 \left (d x +c \right )^{\frac {3}{2}} b^{2} c d e}{3}+\frac {4 \left (d x +c \right )^{\frac {5}{2}} a b d f}{5}-\frac {4 \left (d x +c \right )^{\frac {5}{2}} b^{2} c f}{5}+\frac {2 \left (d x +c \right )^{\frac {5}{2}} b^{2} d e}{5}+\frac {2 \left (d x +c \right )^{\frac {7}{2}} b^{2} f}{7}}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 152, normalized size = 1.04 \begin {gather*} a^{2} \sqrt {c} e \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right ) + \frac {2 \, {\left (105 \, \sqrt {d x + c} a^{2} d^{3} e + 15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} f + 21 \, {\left (b^{2} d e - 2 \, {\left (b^{2} c - a b d\right )} f\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 35 \, {\left ({\left (b^{2} c d - 2 \, a b d^{2}\right )} e - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{105 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.62, size = 263, normalized size = 1.80 \begin {gather*} \left (\frac {2\,b^2\,d\,e-6\,b^2\,c\,f+4\,a\,b\,d\,f}{5\,d^3}+\frac {2\,b^2\,c\,f}{5\,d^3}\right )\,{\left (c+d\,x\right )}^{5/2}+\left (c\,\left (c\,\left (\frac {2\,b^2\,d\,e-6\,b^2\,c\,f+4\,a\,b\,d\,f}{d^3}+\frac {2\,b^2\,c\,f}{d^3}\right )+\frac {2\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-3\,b\,c\,f+2\,b\,d\,e\right )}{d^3}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2\,\left (c\,f-d\,e\right )}{d^3}\right )\,\sqrt {c+d\,x}+\left (\frac {c\,\left (\frac {2\,b^2\,d\,e-6\,b^2\,c\,f+4\,a\,b\,d\,f}{d^3}+\frac {2\,b^2\,c\,f}{d^3}\right )}{3}+\frac {2\,\left (a\,d-b\,c\right )\,\left (a\,d\,f-3\,b\,c\,f+2\,b\,d\,e\right )}{3\,d^3}\right )\,{\left (c+d\,x\right )}^{3/2}+\frac {2\,b^2\,f\,{\left (c+d\,x\right )}^{7/2}}{7\,d^3}+a^2\,\sqrt {c}\,e\,\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.60, size = 167, normalized size = 1.14 \begin {gather*} \frac {2 a^{2} c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + 2 a^{2} e \sqrt {c + d x} + \frac {2 b^{2} f \left (c + d x\right )^{\frac {7}{2}}}{7 d^{3}} + \frac {2 \left (c + d x\right )^{\frac {5}{2}} \left (2 a b d f - 2 b^{2} c f + b^{2} d e\right )}{5 d^{3}} + \frac {2 \left (c + d x\right )^{\frac {3}{2}} \left (a^{2} d^{2} f - 2 a b c d f + 2 a b d^{2} e + b^{2} c^{2} f - b^{2} c d e\right )}{3 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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