Optimal. Leaf size=127 \[ \frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {720, 724, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^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 )}{8 d^{3/2} (c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d (c d-b e)}\\ &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}+\frac {b^2 \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 (c d-b e)}\\ &=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^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 )}{8 d^{3/2} (c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 122, normalized size = 0.96 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {\sqrt {d} (b (d-e x)+2 c d x)}{(d+e x)^2 (c d-b e)}-\frac {b^2 \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x} (c d-b e)^{3/2}}\right )}{4 d^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.02, size = 159, normalized size = 1.25 \begin {gather*} -\frac {b^2 \tanh ^{-1}\left (-\frac {e \sqrt {b x+c x^2}}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {c d-b e}}\right )}{4 d^{3/2} (c d-b e)^{3/2}}-\frac {\sqrt {b x+c x^2} (-b d+b e x-2 c d x)}{4 d (d+e x)^2 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 479, normalized size = 3.77 \begin {gather*} \left [-\frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\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 (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}, -\frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\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 (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 409, normalized size = 3.22 \begin {gather*} -\frac {b^{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 )}{4 \, {\left (c d^{2} - b d e\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 + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} + 8 \, {\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} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} - b^{3} \sqrt {c} d^{2} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2}}{4 \, {\left (c d^{2} e^{2} - b d e^{3}\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1963, normalized size = 15.46
result too large to display
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^3} \,d x \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 {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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