Optimal. Leaf size=126 \[ -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {e^2 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{3/2} (c d-b e)^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {754, 12, 738,
212} \begin {gather*} \frac {e^2 \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 738
Rule 754
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \int -\frac {b^2 e^2}{2 (d+e x) \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {e^2 \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{d (c d-b e)}\\ &=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {e^2 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{3/2} (c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 144, normalized size = 1.14 \begin {gather*} \frac {2 \left (\sqrt {d} \sqrt {-c d+b e} \left (-b^2 e+2 c^2 d x+b c (d-e x)\right )+b^2 e^2 \sqrt {x} \sqrt {b+c x} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )\right )}{b^2 d^{3/2} (-c d+b e)^{3/2} \sqrt {x (b+c x)}} \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. \(339\) vs.
\(2(112)=224\).
time = 0.50, size = 340, normalized size = 2.70
method | result | size |
risch | \(-\frac {2 \left (c x +b \right )}{b^{2} d \sqrt {x \left (c x +b \right )}}+\frac {e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {2 c \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{2} \left (b e -c d \right ) \left (\frac {b}{c}+x \right )}\) | \(214\) |
default | \(\frac {-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e}\) | \(340\) |
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 = 1.67, size = 458, normalized size = 3.63 \begin {gather*} \left [-\frac {{\left (b^{2} c x^{2} + b^{3} x\right )} \sqrt {c d^{2} - b d e} e^{2} \log \left (\frac {2 \, c d x - b x e + b d - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{x e + d}\right ) + 2 \, {\left (2 \, c^{3} d^{3} x + b c^{2} d^{3} + {\left (b^{2} c d x + b^{3} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, b^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}}{b^{2} c^{3} d^{4} x^{2} + b^{3} c^{2} d^{4} x + {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e^{2} - 2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x\right )} e}, \frac {2 \, {\left ({\left (b^{2} c x^{2} + b^{3} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x - b x e}\right ) e^{2} - {\left (2 \, c^{3} d^{3} x + b c^{2} d^{3} + {\left (b^{2} c d x + b^{3} d\right )} e^{2} - {\left (3 \, b c^{2} d^{2} x + 2 \, b^{2} c d^{2}\right )} e\right )} \sqrt {c x^{2} + b x}\right )}}{b^{2} c^{3} d^{4} x^{2} + b^{3} c^{2} d^{4} x + {\left (b^{4} c d^{2} x^{2} + b^{5} d^{2} x\right )} e^{2} - 2 \, {\left (b^{3} c^{2} d^{3} x^{2} + b^{4} c d^{3} x\right )} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 167, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, c^{2} d^{2} - b c d e\right )} x}{b^{2} c d^{3} - b^{3} d^{2} e} + \frac {b c d^{2} - b^{2} d e}{b^{2} c d^{3} - b^{3} d^{2} e}\right )}}{\sqrt {c x^{2} + b x}} - \frac {2 \, \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{2}}{{\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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