Optimal. Leaf size=172 \[ -\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)}{d^4}+\frac {2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}-\frac {2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2} \]
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Rubi [A] time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \begin {gather*} -\frac {2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac {2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)}{d^4}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx &=\int \left (-\frac {b (b d e+b c f-2 a d f) (e+f x)^{3/2}}{d^2 f}+\frac {(-b c+a d)^2 (e+f x)^{3/2}}{d^2 (c+d x)}+\frac {b^2 (e+f x)^{5/2}}{d f}\right ) \, dx\\ &=-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {(b c-a d)^2 \int \frac {(e+f x)^{3/2}}{c+d x} \, dx}{d^2}\\ &=\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left ((b c-a d)^2 (d e-c f)\right ) \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^3}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left ((b c-a d)^2 (d e-c f)^2\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^4}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}+\frac {\left (2 (b c-a d)^2 (d e-c f)^2\right ) \operatorname {Subst}\left (\int \frac {1}{c-\frac {d e}{f}+\frac {d x^2}{f}} \, dx,x,\sqrt {e+f x}\right )}{d^4 f}\\ &=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 160, normalized size = 0.93 \begin {gather*} \frac {2 \left (105 (b c-a d)^2 (d e-c f) \left (\frac {\sqrt {e+f x}}{d}-\frac {\sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )-\frac {21 b d (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{f^2}+35 (e+f x)^{3/2} (b c-a d)^2+\frac {15 b^2 d^2 (e+f x)^{7/2}}{f^2}\right )}{105 d^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 336, normalized size = 1.95 \begin {gather*} -\frac {2 \left (105 a^2 c d^2 f^3 \sqrt {e+f x}-35 a^2 d^3 f^2 (e+f x)^{3/2}-105 a^2 d^3 e f^2 \sqrt {e+f x}-210 a b c^2 d f^3 \sqrt {e+f x}+70 a b c d^2 f^2 (e+f x)^{3/2}+210 a b c d^2 e f^2 \sqrt {e+f x}-42 a b d^3 f (e+f x)^{5/2}+105 b^2 c^3 f^3 \sqrt {e+f x}-35 b^2 c^2 d f^2 (e+f x)^{3/2}-105 b^2 c^2 d e f^2 \sqrt {e+f x}+21 b^2 c d^2 f (e+f x)^{5/2}-15 b^2 d^3 (e+f x)^{7/2}+21 b^2 d^3 e (e+f x)^{5/2}\right )}{105 d^4 f^2}-\frac {2 (a d-b c)^2 (c f-d e)^{3/2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x} \sqrt {c f-d e}}{d e-c f}\right )}{d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.42, size = 694, normalized size = 4.03 \begin {gather*} \left [-\frac {105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (8 \, b^{2} d^{3} e f^{2} - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} + {\left (3 \, b^{2} d^{3} e^{2} f - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}}{105 \, d^{4} f^{2}}, -\frac {2 \, {\left (105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) - {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (8 \, b^{2} d^{3} e f^{2} - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} + {\left (3 \, b^{2} d^{3} e^{2} f - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 424, normalized size = 2.47 \begin {gather*} \frac {2 \, {\left (b^{2} c^{4} f^{2} - 2 \, a b c^{3} d f^{2} + a^{2} c^{2} d^{2} f^{2} - 2 \, b^{2} c^{3} d f e + 4 \, a b c^{2} d^{2} f e - 2 \, a^{2} c d^{3} f e + b^{2} c^{2} d^{2} e^{2} - 2 \, a b c d^{3} e^{2} + a^{2} d^{4} e^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} d^{6} f^{12} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} c d^{5} f^{13} + 42 \, {\left (f x + e\right )}^{\frac {5}{2}} a b d^{6} f^{13} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} d^{4} f^{14} - 70 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c d^{5} f^{14} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{6} f^{14} - 105 \, \sqrt {f x + e} b^{2} c^{3} d^{3} f^{15} + 210 \, \sqrt {f x + e} a b c^{2} d^{4} f^{15} - 105 \, \sqrt {f x + e} a^{2} c d^{5} f^{15} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} d^{6} f^{12} e + 105 \, \sqrt {f x + e} b^{2} c^{2} d^{4} f^{14} e - 210 \, \sqrt {f x + e} a b c d^{5} f^{14} e + 105 \, \sqrt {f x + e} a^{2} d^{6} f^{14} e\right )}}{105 \, d^{7} f^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 644, normalized size = 3.74 \begin {gather*} \frac {2 a^{2} c^{2} f^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{2}}-\frac {4 a^{2} c e f \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d}+\frac {2 a^{2} e^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}-\frac {4 a b \,c^{3} f^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{3}}+\frac {8 a b \,c^{2} e f \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{2}}-\frac {4 a b c \,e^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d}+\frac {2 b^{2} c^{4} f^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{4}}-\frac {4 b^{2} c^{3} e f \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{3}}+\frac {2 b^{2} c^{2} e^{2} \arctan \left (\frac {\sqrt {f x +e}\, d}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}\, d^{2}}-\frac {2 \sqrt {f x +e}\, a^{2} c f}{d^{2}}+\frac {2 \sqrt {f x +e}\, a^{2} e}{d}+\frac {4 \sqrt {f x +e}\, a b \,c^{2} f}{d^{3}}-\frac {4 \sqrt {f x +e}\, a b c e}{d^{2}}-\frac {2 \sqrt {f x +e}\, b^{2} c^{3} f}{d^{4}}+\frac {2 \sqrt {f x +e}\, b^{2} c^{2} e}{d^{3}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} a^{2}}{3 d}-\frac {4 \left (f x +e \right )^{\frac {3}{2}} a b c}{3 d^{2}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{2} c^{2}}{3 d^{3}}+\frac {4 \left (f x +e \right )^{\frac {5}{2}} a b}{5 d f}-\frac {2 \left (f x +e \right )^{\frac {5}{2}} b^{2} c}{5 d^{2} f}-\frac {2 \left (f x +e \right )^{\frac {5}{2}} b^{2} e}{5 d \,f^{2}}+\frac {2 \left (f x +e \right )^{\frac {7}{2}} b^{2}}{7 d \,f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 438, normalized size = 2.55 \begin {gather*} {\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{3\,d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d\,f^2}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{5\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{a^2\,c^2\,d^2\,f^2-2\,a^2\,c\,d^3\,e\,f+a^2\,d^4\,e^2-2\,a\,b\,c^3\,d\,f^2+4\,a\,b\,c^2\,d^2\,e\,f-2\,a\,b\,c\,d^3\,e^2+b^2\,c^4\,f^2-2\,b^2\,c^3\,d\,e\,f+b^2\,c^2\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{9/2}}-\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.19, size = 236, normalized size = 1.37 \begin {gather*} \frac {2 b^{2} \left (e + f x\right )^{\frac {7}{2}}}{7 d f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (4 a b d f - 2 b^{2} c f - 2 b^{2} d e\right )}{5 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (2 a^{2} d^{2} - 4 a b c d + 2 b^{2} c^{2}\right )}{3 d^{3}} + \frac {\sqrt {e + f x} \left (- 2 a^{2} c d^{2} f + 2 a^{2} d^{3} e + 4 a b c^{2} d f - 4 a b c d^{2} e - 2 b^{2} c^{3} f + 2 b^{2} c^{2} d e\right )}{d^{4}} + \frac {2 \left (a d - b c\right )^{2} \left (c f - d e\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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