Optimal. Leaf size=170 \[ -\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (3 b d e-2 b c f-a d f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 162, 65,
214, 211} \begin {gather*} \frac {2 (d e-c f)^{3/2} \text {ArcTan}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (-a d f-2 b c f+3 b d e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}-\frac {\sqrt {c+d x} (b c-a d)}{b (a+b x) (b e-a f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 162
Rule 211
Rule 214
Rubi steps
\begin {align*} \int \frac {(c+d x)^{3/2}}{(a+b x)^2 (e+f x)} \, dx &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}-\frac {\int \frac {\frac {1}{2} \left (a d^2 e-2 b c \left (\frac {3 d e}{2}-c f\right )\right )-\frac {1}{2} d (2 b d e-b c f-a d f) x}{(a+b x) \sqrt {c+d x} (e+f x)} \, dx}{b (b e-a f)}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {(d e-c f)^2 \int \frac {1}{\sqrt {c+d x} (e+f x)} \, dx}{(b e-a f)^2}+\frac {((b c-a d) (3 b d e-2 b c f-a d f)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b (b e-a f)^2}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {\left (2 (d e-c f)^2\right ) \text {Subst}\left (\int \frac {1}{e-\frac {c f}{d}+\frac {f x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{d (b e-a f)^2}+\frac {((b c-a d) (3 b d e-2 b c f-a d f)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{b d (b e-a f)^2}\\ &=-\frac {(b c-a d) \sqrt {c+d x}}{b (b e-a f) (a+b x)}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f} (b e-a f)^2}-\frac {\sqrt {b c-a d} (3 b d e-2 b c f-a d f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{3/2} (b e-a f)^2}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 156, normalized size = 0.92 \begin {gather*} \frac {\frac {(-b c+a d) (b e-a f) \sqrt {c+d x}}{b (a+b x)}+\frac {\sqrt {-b c+a d} (-3 b d e+2 b c f+a d f) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2}}+\frac {2 (d e-c f)^{3/2} \tan ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d e-c f}}\right )}{\sqrt {f}}}{(b e-a f)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 181, normalized size = 1.06
method | result | size |
derivativedivides | \(2 d^{2} \left (-\frac {\left (c f -d e \right )^{2} \arctanh \left (\frac {f \sqrt {d x +c}}{\sqrt {\left (c f -d e \right ) f}}\right )}{d^{2} \left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}+\frac {\left (a d -b c \right ) \left (-\frac {d \left (a f -b e \right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d f +2 b c f -3 b d e \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a f -b e \right )^{2}}\right )\) | \(181\) |
default | \(2 d^{2} \left (-\frac {\left (c f -d e \right )^{2} \arctanh \left (\frac {f \sqrt {d x +c}}{\sqrt {\left (c f -d e \right ) f}}\right )}{d^{2} \left (a f -b e \right )^{2} \sqrt {\left (c f -d e \right ) f}}+\frac {\left (a d -b c \right ) \left (-\frac {d \left (a f -b e \right ) \sqrt {d x +c}}{2 b \left (b \left (d x +c \right )+a d -b c \right )}+\frac {\left (a d f +2 b c f -3 b d e \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a f -b e \right )^{2}}\right )\) | \(181\) |
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 = 2.63, size = 1139, normalized size = 6.70 \begin {gather*} \left [-\frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {c f - d e}{f}} \log \left (\frac {d f x + 2 \, c f + 2 \, \sqrt {d x + c} f \sqrt {\frac {c f - d e}{f}} - d e}{f x + e}\right ) - 2 \, {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e\right )}}, -\frac {4 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {c f - d e}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {-\frac {c f - d e}{f}}}{c f - d e}\right ) + {\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) - 2 \, {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{2 \, {\left (a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e\right )}}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {\frac {c f - d e}{f}} \log \left (\frac {d f x + 2 \, c f + 2 \, \sqrt {d x + c} f \sqrt {\frac {c f - d e}{f}} - d e}{f x + e}\right ) + {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e}, \frac {{\left ({\left (2 \, b^{2} c + a b d\right )} f x + {\left (2 \, a b c + a^{2} d\right )} f - 3 \, {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - 2 \, {\left (b^{2} c f x + a b c f - {\left (b^{2} d x + a b d\right )} e\right )} \sqrt {-\frac {c f - d e}{f}} \arctan \left (-\frac {\sqrt {d x + c} f \sqrt {-\frac {c f - d e}{f}}}{c f - d e}\right ) + {\left ({\left (a b c - a^{2} d\right )} f - {\left (b^{2} c - a b d\right )} e\right )} \sqrt {d x + c}}{a^{2} b^{2} f^{2} x + a^{3} b f^{2} + {\left (b^{4} x + a b^{3}\right )} e^{2} - 2 \, {\left (a b^{3} f x + a^{2} b^{2} f\right )} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1766 vs.
\(2 (146) = 292\).
time = 143.95, size = 1766, normalized size = 10.39 \begin {gather*} \frac {a^{2} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a b f - 2 b^{2} e} - \frac {a^{2} d^{3} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a b f - 2 b^{2} e} - \frac {2 a^{2} d^{3} \sqrt {c + d x}}{2 a^{3} b d^{2} f - 2 a^{2} b^{2} c d f - 2 a^{2} b^{2} d^{2} e + 2 a^{2} b^{2} d^{2} f x + 2 a b^{3} c d e - 2 a b^{3} c d f x - 2 a b^{3} d^{2} e x + 2 b^{4} c d e x} + \frac {2 a^{2} d^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b^{2} \left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} - \frac {a b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a b f - 2 b^{2} e} + \frac {a b c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a b f - 2 b^{2} e} + \frac {2 a b c d^{2} \sqrt {c + d x}}{2 a^{3} b d^{2} f - 2 a^{2} b^{2} c d f - 2 a^{2} b^{2} d^{2} e + 2 a^{2} b^{2} d^{2} f x + 2 a b^{3} c d e - 2 a b^{3} c d f x - 2 a b^{3} d^{2} e x + 2 b^{4} c d e x} - \frac {a c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a f - 2 b e} + \frac {a c d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a f - 2 b e} + \frac {2 a c d^{2} \sqrt {c + d x}}{2 a^{3} d^{2} f - 2 a^{2} b c d f - 2 a^{2} b d^{2} e + 2 a^{2} b d^{2} f x + 2 a b^{2} c d e - 2 a b^{2} c d f x - 2 a b^{2} d^{2} e x + 2 b^{3} c d e x} - \frac {4 a d^{2} e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{b \left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} + \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (- a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a f - 2 b e} - \frac {b c^{2} d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} \log {\left (a^{2} d^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} - 2 a b c d \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + b^{2} c^{2} \sqrt {- \frac {1}{b \left (a d - b c\right )^{3}}} + \sqrt {c + d x} \right )}}{2 a f - 2 b e} - \frac {2 b c^{2} d \sqrt {c + d x}}{2 a^{3} d^{2} f - 2 a^{2} b c d f - 2 a^{2} b d^{2} e + 2 a^{2} b d^{2} f x + 2 a b^{2} c d e - 2 a b^{2} c d f x - 2 a b^{2} d^{2} e x + 2 b^{3} c d e x} - \frac {2 c^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{\left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} + \frac {2 c^{2} f \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{\sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} + \frac {4 c d e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d}{b} - c}} \right )}}{\left (a f - b e\right )^{2} \sqrt {\frac {a d}{b} - c}} - \frac {4 c d e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{\sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} + \frac {2 d^{2} e^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + \frac {d e}{f}}} \right )}}{f \sqrt {- c + \frac {d e}{f}} \left (a f - b e\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 250, normalized size = 1.47 \begin {gather*} -\frac {{\left (2 \, b^{2} c^{2} f - a b c d f - a^{2} d^{2} f - 3 \, b^{2} c d e + 3 \, a b d^{2} e\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{2} b f^{2} - 2 \, a b^{2} f e + b^{3} e^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (c^{2} f^{2} - 2 \, c d f e + d^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {d x + c} f}{\sqrt {-c f^{2} + d f e}}\right )}{{\left (a^{2} f^{2} - 2 \, a b f e + b^{2} e^{2}\right )} \sqrt {-c f^{2} + d f e}} + \frac {\sqrt {d x + c} b c d - \sqrt {d x + c} a d^{2}}{{\left (a b f - b^{2} e\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.82, size = 2500, normalized size = 14.71 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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