Optimal. Leaf size=210 \[ \frac {2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac {2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3}{d^4}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
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Rubi [A] time = 0.18, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 50, 63, 208} \[ \frac {2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac {2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}-\frac {2 \sqrt {e+f x} (b c-a d)^3}{d^4}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
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)^3 \sqrt {e+f x}}{c+d x} \, dx &=\int \left (\frac {b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) \sqrt {e+f x}}{d^3 f^2}+\frac {(-b c+a d)^3 \sqrt {e+f x}}{d^3 (c+d x)}-\frac {b^2 (2 b d e+b c f-3 a d f) (e+f x)^{3/2}}{d^2 f^2}+\frac {b^3 (e+f x)^{5/2}}{d f^2}\right ) \, dx\\ &=\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac {(b c-a d)^3 \int \frac {\sqrt {e+f x}}{c+d x} \, dx}{d^3}\\ &=-\frac {2 (b c-a d)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac {\left ((b c-a d)^3 (d e-c f)\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}} \, dx}{d^4}\\ &=-\frac {2 (b c-a d)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}-\frac {\left (2 (b c-a d)^3 (d e-c f)\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)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \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 = 0.28, size = 210, normalized size = 1.00 \[ \frac {2 b (e+f x)^{3/2} \left (3 a^2 d^2 f^2-3 a b d f (c f+d e)+b^2 \left (c^2 f^2+c d e f+d^2 e^2\right )\right )}{3 d^3 f^3}-\frac {2 b^2 (e+f x)^{5/2} (-3 a d f+b c f+2 b d e)}{5 d^2 f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}-\frac {2 \sqrt {e+f x} (b c-a d)^3}{d^4}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 705, normalized size = 3.36 \[ \left [-\frac {105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \, {\left (b^{3} d^{3} e f^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{3} e^{2} f + 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}}{105 \, d^{4} f^{3}}, \frac {2 \, {\left (105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) + {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} + 14 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e^{2} f + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} e f^{2} - 105 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} + 3 \, {\left (b^{3} d^{3} e f^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3}\right )} x^{2} - {\left (4 \, b^{3} d^{3} e^{2} f + 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} e f^{2} - 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.34, size = 436, normalized size = 2.08 \[ \frac {2 \, {\left (b^{3} c^{4} f - 3 \, a b^{2} c^{3} d f + 3 \, a^{2} b c^{2} d^{2} f - a^{3} c d^{3} f - b^{3} c^{3} d e + 3 \, a b^{2} c^{2} d^{2} e - 3 \, a^{2} b c d^{3} e + a^{3} d^{4} e\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {c d f - d^{2} e}}\right )}{\sqrt {c d f - d^{2} e} d^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{3} d^{6} f^{18} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} c d^{5} f^{19} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} a b^{2} d^{6} f^{19} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} f^{20} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c d^{5} f^{20} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d^{6} f^{20} - 105 \, \sqrt {f x + e} b^{3} c^{3} d^{3} f^{21} + 315 \, \sqrt {f x + e} a b^{2} c^{2} d^{4} f^{21} - 315 \, \sqrt {f x + e} a^{2} b c d^{5} f^{21} + 105 \, \sqrt {f x + e} a^{3} d^{6} f^{21} - 42 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{3} d^{6} f^{18} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c d^{5} f^{19} e - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d^{6} f^{19} e + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d^{6} f^{18} e^{2}\right )}}{105 \, d^{7} f^{21}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 629, normalized size = 3.00 \[ -\frac {2 a^{3} c 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^{3} e \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 {6 a^{2} b \,c^{2} 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 {6 a^{2} b c e \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 {6 a \,b^{2} c^{3} 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 {6 a \,b^{2} c^{2} e \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 b^{3} c^{4} 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^{4}}-\frac {2 b^{3} c^{3} e \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 \sqrt {f x +e}\, a^{3}}{d}-\frac {6 \sqrt {f x +e}\, a^{2} b c}{d^{2}}+\frac {6 \sqrt {f x +e}\, a \,b^{2} c^{2}}{d^{3}}-\frac {2 \sqrt {f x +e}\, b^{3} c^{3}}{d^{4}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} a^{2} b}{d f}-\frac {2 \left (f x +e \right )^{\frac {3}{2}} a \,b^{2} c}{d^{2} f}-\frac {2 \left (f x +e \right )^{\frac {3}{2}} a \,b^{2} e}{d \,f^{2}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{3} c^{2}}{3 d^{3} f}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{3} c e}{3 d^{2} f^{2}}+\frac {2 \left (f x +e \right )^{\frac {3}{2}} b^{3} e^{2}}{3 d \,f^{3}}+\frac {6 \left (f x +e \right )^{\frac {5}{2}} a \,b^{2}}{5 d \,f^{2}}-\frac {2 \left (f x +e \right )^{\frac {5}{2}} b^{3} c}{5 d^{2} f^{2}}-\frac {4 \left (f x +e \right )^{\frac {5}{2}} b^{3} e}{5 d \,f^{3}}+\frac {2 \left (f x +e \right )^{\frac {7}{2}} b^{3}}{7 d \,f^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 451, normalized size = 2.15 \[ {\left (e+f\,x\right )}^{3/2}\,\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{3\,d\,f^3}+\frac {2\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{5\,d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{5\,d^2\,f^6}\right )+\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^3}{d\,f^3}-\frac {\left (\frac {\left (\frac {6\,b^3\,e-6\,a\,b^2\,f}{d\,f^3}+\frac {2\,b^3\,\left (c\,f^4-d\,e\,f^3\right )}{d^2\,f^6}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}+\frac {6\,b\,{\left (a\,f-b\,e\right )}^2}{d\,f^3}\right )\,\left (c\,f^4-d\,e\,f^3\right )}{d\,f^3}\right )+\frac {2\,b^3\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,1{}\mathrm {i}}{-f\,a^3\,c\,d^3+e\,a^3\,d^4+3\,f\,a^2\,b\,c^2\,d^2-3\,e\,a^2\,b\,c\,d^3-3\,f\,a\,b^2\,c^3\,d+3\,e\,a\,b^2\,c^2\,d^2+f\,b^3\,c^4-e\,b^3\,c^3\,d}\right )\,{\left (a\,d-b\,c\right )}^3\,\sqrt {d\,e-c\,f}\,2{}\mathrm {i}}{d^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.41, size = 269, normalized size = 1.28 \[ \frac {2 \left (\frac {b^{3} \left (e + f x\right )^{\frac {7}{2}}}{7 d f^{2}} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (3 a b^{2} d f - b^{3} c f - 2 b^{3} d e\right )}{5 d^{2} f^{2}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (3 a^{2} b d^{2} f^{2} - 3 a b^{2} c d f^{2} - 3 a b^{2} d^{2} e f + b^{3} c^{2} f^{2} + b^{3} c d e f + b^{3} d^{2} e^{2}\right )}{3 d^{3} f^{2}} + \frac {\sqrt {e + f x} \left (a^{3} d^{3} f - 3 a^{2} b c d^{2} f + 3 a b^{2} c^{2} d f - b^{3} c^{3} f\right )}{d^{4}} - \frac {f \left (a d - b c\right )^{3} \left (c f - d e\right ) \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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