Optimal. Leaf size=210 \[ -\frac {2 (b c-a d)^3 \sqrt {e+f x}}{d^4}+\frac {2 b \left (3 a^2 d^2 f^2-3 a b d f (d e+c f)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right )\right ) (e+f x)^{3/2}}{3 d^3 f^3}-\frac {2 b^2 (2 b d e+b c f-3 a d f) (e+f x)^{5/2}}{5 d^2 f^3}+\frac {2 b^3 (e+f x)^{7/2}}{7 d f^3}+\frac {2 (b c-a d)^3 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \]
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Rubi [A]
time = 0.10, 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 = {90, 52, 65, 214}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 214
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 ) \text {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 = 0.37, size = 249, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {e+f x} \left (105 a^3 d^3 f^3+105 a^2 b d^2 f^2 (-3 c f+d (e+f x))-21 a b^2 d f \left (-15 c^2 f^2+5 c d f (e+f x)+d^2 \left (2 e^2-e f x-3 f^2 x^2\right )\right )+b^3 \left (-105 c^3 f^3+35 c^2 d f^2 (e+f x)-7 c d^2 f \left (-2 e^2+e f x+3 f^2 x^2\right )+d^3 \left (8 e^3-4 e^2 f x+3 e f^2 x^2+15 f^3 x^3\right )\right )\right )}{105 d^4 f^3}-\frac {2 (-b c+a d)^3 \sqrt {-d e+c f} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(389\) vs.
\(2(186)=372\).
time = 0.10, size = 390, normalized size = 1.86
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} d^{4} e -3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(390\) |
default | \(\frac {\frac {2 \left (\frac {b^{3} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {3 a \,b^{2} d^{3} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {b^{3} c \,d^{2} f \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} d^{3} e \left (f x +e \right )^{\frac {5}{2}}}{5}+a^{2} b \,d^{3} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} c \,d^{2} f^{2} \left (f x +e \right )^{\frac {3}{2}}-a \,b^{2} d^{3} e f \left (f x +e \right )^{\frac {3}{2}}+\frac {b^{3} c^{2} d \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} c \,d^{2} e f \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{3} d^{3} e^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{3} d^{3} f^{3} \sqrt {f x +e}-3 a^{2} b c \,d^{2} f^{3} \sqrt {f x +e}+3 a \,b^{2} c^{2} d \,f^{3} \sqrt {f x +e}-b^{3} c^{3} f^{3} \sqrt {f x +e}\right )}{d^{4}}-\frac {2 f^{3} \left (a^{3} c \,d^{3} f -a^{3} d^{4} e -3 a^{2} b \,c^{2} d^{2} f +3 a^{2} b c \,d^{3} e +3 a \,b^{2} c^{3} d f -3 a \,b^{2} c^{2} d^{2} e -b^{3} c^{4} f +b^{3} c^{3} d e \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{3}}\) | \(390\) |
risch | \(\frac {2 \left (15 b^{3} d^{3} f^{3} x^{3}+63 a \,b^{2} d^{3} f^{3} x^{2}-21 b^{3} c \,d^{2} f^{3} x^{2}+3 b^{3} d^{3} e \,f^{2} x^{2}+105 a^{2} b \,d^{3} f^{3} x -105 a \,b^{2} c \,d^{2} f^{3} x +21 a \,b^{2} d^{3} e \,f^{2} x +35 b^{3} c^{2} d \,f^{3} x -7 b^{3} c \,d^{2} e \,f^{2} x -4 b^{3} d^{3} e^{2} f x +105 a^{3} d^{3} f^{3}-315 a^{2} b c \,d^{2} f^{3}+105 a^{2} b \,d^{3} e \,f^{2}+315 a \,b^{2} c^{2} d \,f^{3}-105 a \,b^{2} c \,d^{2} e \,f^{2}-42 a \,b^{2} d^{3} e^{2} f -105 b^{3} c^{3} f^{3}+35 b^{3} c^{2} d e \,f^{2}+14 b^{3} c \,d^{2} e^{2} f +8 b^{3} d^{3} e^{3}\right ) \sqrt {f x +e}}{105 f^{3} d^{4}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3} c f}{d \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{3} e}{\sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b \,c^{2} f}{d^{2} \sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a^{2} b c e}{d \sqrt {\left (c f -d e \right ) d}}-\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{3} f}{d^{3} \sqrt {\left (c f -d e \right ) d}}+\frac {6 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) a \,b^{2} c^{2} e}{d^{2} \sqrt {\left (c f -d e \right ) d}}+\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{4} f}{d^{4} \sqrt {\left (c f -d e \right ) d}}-\frac {2 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) b^{3} c^{3} e}{d^{3} \sqrt {\left (c f -d e \right ) d}}\) | \(645\) |
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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Fricas [A]
time = 1.44, size = 714, normalized size = 3.40 \begin {gather*} \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 {c f - d e}{d}} \log \left (\frac {d f x - c f - 2 \, \sqrt {f x + e} d \sqrt {-\frac {c f - d e}{d}} + 2 \, d e}{d x + c}\right ) - 2 \, {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} - 21 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3} x^{2} + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3} x - 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} - 2 \, {\left (2 \, b^{3} d^{3} f x - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} e^{2} + {\left (3 \, b^{3} d^{3} f^{2} x^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{2} x + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{2}\right )} e\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 {c f - d e}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {\frac {c f - d e}{d}}}{c f - d e}\right ) - {\left (15 \, b^{3} d^{3} f^{3} x^{3} + 8 \, b^{3} d^{3} e^{3} - 21 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{3} x^{2} + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{3} x - 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} - 2 \, {\left (2 \, b^{3} d^{3} f x - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f\right )} e^{2} + {\left (3 \, b^{3} d^{3} f^{2} x^{2} - 7 \, {\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} f^{2} x + 35 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} f^{2}\right )} e\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.05, size = 269, normalized size = 1.28 \begin {gather*} \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}} \cdot \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}} \cdot \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 436 vs.
\(2 (195) = 390\).
time = 0.51, size = 436, normalized size = 2.08 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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