Optimal. Leaf size=180 \[ \frac {\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac {3 e \sqrt {b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac {e \sqrt {b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]
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Rubi [A] time = 0.17, antiderivative size = 180, 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 = {744, 806, 724, 206} \[ \frac {\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac {3 e \sqrt {b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac {e \sqrt {b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 744
Rule 806
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} (-4 c d+3 b e)+c e x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d^2 (c d-b e)^2}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}-\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \operatorname {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 )}{4 d^2 (c d-b e)^2}\\ &=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \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 )}{8 d^{5/2} (c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 183, normalized size = 1.02 \[ \frac {\sqrt {x} \left (-\frac {\sqrt {b+c x} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}+\frac {3 e \sqrt {x} (b+c x) (2 c d-b e)}{2 d (d+e x) (c d-b e)}+\frac {e \sqrt {x} (b+c x)}{(d+e x)^2}\right )}{2 d \sqrt {x (b+c x)} (b e-c d)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 743, normalized size = 4.13 \[ \left [\frac {{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + {\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \, {\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}, \frac {{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + {\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \, {\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 487, normalized size = 2.71 \[ -\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} - \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 6 \, b^{2} c^{\frac {3}{2}} d^{3} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 3 \, b^{3} \sqrt {c} d^{2} e + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 798, normalized size = 4.43 \[ -\frac {3 b^{2} e \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{8 \left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d^{2}}+\frac {3 b c \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{2 \left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d}-\frac {3 c^{2} \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{2 \left (b e -c d \right )^{2} \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e}+\frac {3 \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, b e}{4 \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right ) d^{2}}-\frac {c \ln \left (\frac {-\frac {2 \left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{x +\frac {d}{e}}\right )}{2 \left (b e -c d \right ) \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, d e}-\frac {3 \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}\, c}{2 \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right ) d}+\frac {\sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {\left (b e -c d \right ) d}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}}}{2 \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2} d e} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {1}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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