Optimal. Leaf size=216 \[ \frac {\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac {\sqrt {b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{4 d^2 (d+e x) (c d-b e)^2}+\frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)} \]
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Rubi [A] time = 0.26, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {834, 806, 724, 206} \begin {gather*} \frac {\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A 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 {\sqrt {b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{4 d^2 (d+e x) (c d-b e)^2}+\frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx &=\frac {(B d-A e) \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} (b B d-4 A c d+3 A b e)-c (B d-A e) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{2 d (c d-b e)}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {(3 A e (2 c d-b e)-B d (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 A c^2 d^2-4 b c d (B d+2 A e)+b^2 e (B d+3 A e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d^2 (c d-b e)^2}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {(3 A e (2 c d-b e)-B d (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 A c^2 d^2-4 b c d (B d+2 A e)+b^2 e (B d+3 A e)\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 {(B d-A e) \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {(3 A e (2 c d-b e)-B d (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 A c^2 d^2-4 b c d (B d+2 A e)+b^2 e (B d+3 A e)\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 = 0.39, size = 217, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x} \left (-\frac {\sqrt {b+c x} \left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A 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 {\sqrt {x} (b+c x) (3 A e (b e-2 c d)+B d (b e+2 c d))}{2 d (d+e x) (c d-b e)}+\frac {\sqrt {x} (b+c x) (A e-B d)}{(d+e x)^2}\right )}{2 d \sqrt {x (b+c x)} (b e-c d)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.03, size = 217, normalized size = 1.00 \begin {gather*} \frac {\left (3 A b^2 e^2-8 A b c d e+8 A c^2 d^2+b^2 B d e-4 b B c d^2\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{4 d^{5/2} (c d-b e)^{5/2}}+\frac {\sqrt {b x+c x^2} \left (5 A b d e^2+3 A b e^3 x-8 A c d^2 e-6 A c d e^2 x-b B d^2 e+b B d e^2 x+4 B c d^3+2 B c d^2 e x\right )}{4 d^2 (d+e x)^2 (c d-b e)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 954, normalized size = 4.42 \begin {gather*} \left [\frac {{\left (3 \, A b^{2} d^{2} e^{2} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + {\left (B b^{2} - 8 \, A b c\right )} d^{3} e + {\left (3 \, A b^{2} e^{4} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} e^{2} + {\left (B b^{2} - 8 \, A b c\right )} d e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{2} d e^{3} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{3} e + {\left (B b^{2} - 8 \, A b c\right )} d^{2} e^{2}\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 (4 \, B c^{2} d^{5} - 5 \, A b^{2} d^{2} e^{3} - {\left (5 \, B b c + 8 \, A c^{2}\right )} d^{4} e + {\left (B b^{2} + 13 \, A b c\right )} d^{3} e^{2} + {\left (2 \, B c^{2} d^{4} e - 3 \, A b^{2} d e^{4} - {\left (B b c + 6 \, A c^{2}\right )} d^{3} e^{2} - {\left (B b^{2} - 9 \, A b c\right )} d^{2} e^{3}\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 (3 \, A b^{2} d^{2} e^{2} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + {\left (B b^{2} - 8 \, A b c\right )} d^{3} e + {\left (3 \, A b^{2} e^{4} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} e^{2} + {\left (B b^{2} - 8 \, A b c\right )} d e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{2} d e^{3} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{3} e + {\left (B b^{2} - 8 \, A b c\right )} d^{2} e^{2}\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 (4 \, B c^{2} d^{5} - 5 \, A b^{2} d^{2} e^{3} - {\left (5 \, B b c + 8 \, A c^{2}\right )} d^{4} e + {\left (B b^{2} + 13 \, A b c\right )} d^{3} e^{2} + {\left (2 \, B c^{2} d^{4} e - 3 \, A b^{2} d e^{4} - {\left (B b c + 6 \, A c^{2}\right )} d^{3} e^{2} - {\left (B b^{2} - 9 \, A b c\right )} d^{2} e^{3}\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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 769, normalized size = 3.56 \begin {gather*} -\frac {{\left (4 \, B b c d^{2} - 8 \, A c^{2} d^{2} - B b^{2} d e + 8 \, A b c d e - 3 \, A 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 )}^{2} B c^{\frac {5}{2}} d^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{\frac {3}{2}} d^{3} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{\frac {5}{2}} d^{3} e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{2} d^{4} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c d^{2} e^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{2} d^{2} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{2} d^{3} e + 2 \, B b^{2} c^{\frac {3}{2}} d^{4} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} \sqrt {c} d^{2} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{\frac {3}{2}} d^{2} e^{2} + B b^{3} \sqrt {c} d^{3} e - 6 \, A b^{2} c^{\frac {3}{2}} d^{3} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} d e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c d e^{3} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} d^{2} e^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c d^{2} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} \sqrt {c} d e^{3} + 3 \, A b^{3} \sqrt {c} d^{2} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} e^{4} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} d e^{3}}{4 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} 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 = 0.06, size = 1821, normalized size = 8.43
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 = -0.00 \begin {gather*} \int \frac {A+B\,x}{\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 {A + B x}{\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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