Optimal. Leaf size=128 \[ \frac {\sqrt {b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac {(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {806, 724, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac {(A b e-2 A c d+b B d) \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 )}{2 d^{3/2} (c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx &=\frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}+\frac {(b B d-2 A c d+A b e) \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 )}{d (c d-b e)}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{d (c d-b e) (d+e x)}-\frac {(b B d-2 A c d+A b e) \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 )}{2 d^{3/2} (c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 133, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x} \left (\frac {\sqrt {d} \sqrt {x} (b+c x) (A e-B d)}{d+e x}+\frac {\sqrt {b+c x} (A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {c d-b e}}\right )}{d^{3/2} \sqrt {x (b+c x)} (b e-c d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.95, size = 930, normalized size = 7.27 \begin {gather*} \frac {B d b^3+B e x b^3+c^{5/2} \sqrt {c x^2+b x} \left (8 B d x^2+8 A d x\right )+c^{3/2} \sqrt {c x^2+b x} \left (-8 b B e x^2-4 A b e x+4 A b d\right )+c \left (8 B e x^2 b^2-A d b^2+4 B d x b^2+3 A e x b^2\right )+c^3 \left (-8 B d x^3-8 A d x^2\right )+c^2 \left (8 b B e x^3-4 b B d x^2+4 A b e x^2-8 A b d x\right )+\sqrt {c} \left (-3 B d b^2-A e b^2-4 B e x b^2\right ) \sqrt {c x^2+b x}}{8 d^2 x^2 (d+e x) c^{7/2}-8 d^2 x (d+e x) \sqrt {c x^2+b x} c^3+d (d+e x) \left (8 b d x-8 b e x^2\right ) c^{5/2}+d (d+e x) (8 b e x-4 b d) \sqrt {c x^2+b x} c^2+d (d+e x) \left (b^2 d-8 b^2 e x\right ) c^{3/2}+4 b^2 d e (d+e x) \sqrt {c x^2+b x} c-b^3 d e (d+e x) \sqrt {c}}+\frac {2 b B \tan ^{-1}\left (\frac {\sqrt {c} x e}{\sqrt {d} \sqrt {b e-c d}}-\frac {\sqrt {c x^2+b x} e}{\sqrt {d} \sqrt {b e-c d}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {b e-c d}}\right )}{\sqrt {d} (c d-b e) \sqrt {b e-c d}}-\frac {2 B c \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {c} x e}{\sqrt {d} \sqrt {b e-c d}}-\frac {\sqrt {c x^2+b x} e}{\sqrt {d} \sqrt {b e-c d}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {b e-c d}}\right )}{e (c d-b e) \sqrt {b e-c d}}+\left (\frac {b B}{\sqrt {d} (c d-b e)^{3/2}}+\frac {2 A c}{\sqrt {d} (c d-b e)^{3/2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {c} d+\sqrt {c} e x-e \sqrt {c x^2+b x}}{\sqrt {d} \sqrt {c d-b e}}\right )-\frac {A b e \tanh ^{-1}\left (\frac {\sqrt {c} x e}{\sqrt {d} \sqrt {c d-b e}}-\frac {\sqrt {c x^2+b x} e}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac {2 B c \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {c} x e}{\sqrt {d} \sqrt {c d-b e}}-\frac {\sqrt {c x^2+b x} e}{\sqrt {d} \sqrt {c d-b e}}+\frac {\sqrt {c} \sqrt {d}}{\sqrt {c d-b e}}\right )}{e (c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 399, normalized size = 3.12 \begin {gather*} \left [-\frac {{\left (A b d e + {\left (B b - 2 \, A c\right )} d^{2} + {\left (A b e^{2} + {\left (B b - 2 \, A c\right )} d e\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 (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {c x^{2} + b x}}{2 \, {\left (c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x\right )}}, -\frac {{\left (A b d e + {\left (B b - 2 \, A c\right )} d^{2} + {\left (A b e^{2} + {\left (B b - 2 \, A c\right )} d e\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 (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} \sqrt {c x^{2} + b x}}{c^{2} d^{5} - 2 \, b c d^{4} e + b^{2} d^{3} e^{2} + {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.10, size = 509, normalized size = 3.98 \begin {gather*} -\frac {1}{2} \, {\left (\frac {{\left (B b d e^{2} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) - 2 \, A c d e^{2} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) + 2 \, \sqrt {c d^{2} - b d e} B \sqrt {c} d e + A b e^{3} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} \right |}\right ) - 2 \, \sqrt {c d^{2} - b d e} A \sqrt {c} e^{2}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c d^{2} - b d e} c d^{2} - \sqrt {c d^{2} - b d e} b d e} - \frac {2 \, {\left (B d e \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - A e^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} \sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}}}{c d^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{2} - b d e \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{2}} - \frac {{\left (B b d e^{3} - 2 \, A c d e^{3} + A b e^{4}\right )} \log \left ({\left | 2 \, c d - b e - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{{\left (c d^{2} e - b d e^{2}\right )} \sqrt {c d^{2} - b d e} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 849, normalized size = 6.63 \begin {gather*} -\frac {A b \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}+\frac {A 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 )}{\left (b e -c d \right ) \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e}+\frac {B b \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}}}\, e}-\frac {B c d \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 )}{\left (b e -c d \right ) \sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e^{2}}+\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}}\, A}{\left (b e -c d \right ) \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}}\, B}{\left (b e -c d \right ) \left (x +\frac {d}{e}\right ) e}-\frac {B \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 )}{\sqrt {-\frac {\left (b e -c d \right ) d}{e^{2}}}\, e^{2}} \end {gather*}
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 {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^2} \,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 {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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