Optimal. Leaf size=85 \[ \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}-\frac {2 \sqrt {d+e x} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {78, 63, 217, 206} \[ \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}-\frac {2 \sqrt {d+e x} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{3/2} \sqrt {d+e x}} \, dx &=-\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b x}}+\frac {B \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{b}\\ &=-\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2}\\ &=-\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b x}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b^2}\\ &=-\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 117, normalized size = 1.38 \[ \frac {2 \left (\frac {b (d+e x) (a B-A b)}{\sqrt {a+b x} (b d-a e)}+\frac {B \sqrt {b d-a e} \sqrt {\frac {b (d+e x)}{b d-a e}} \sinh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b d-a e}}\right )}{\sqrt {e}}\right )}{b^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.46, size = 360, normalized size = 4.24 \[ \left [\frac {4 \, {\left (B a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} e + {\left (B a b d - B a^{2} e + {\left (B b^{2} d - B a b e\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right )}{2 \, {\left (a b^{3} d e - a^{2} b^{2} e^{2} + {\left (b^{4} d e - a b^{3} e^{2}\right )} x\right )}}, \frac {2 \, {\left (B a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} e - {\left (B a b d - B a^{2} e + {\left (B b^{2} d - B a b e\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right )}{a b^{3} d e - a^{2} b^{2} e^{2} + {\left (b^{4} d e - a b^{3} e^{2}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 135, normalized size = 1.59 \[ -\frac {B e^{\left (-\frac {1}{2}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}{\sqrt {b} {\left | b \right |}} + \frac {4 \, {\left (B a \sqrt {b} e^{\frac {1}{2}} - A b^{\frac {3}{2}} e^{\frac {1}{2}}\right )}}{{\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 278, normalized size = 3.27 \[ \frac {\sqrt {e x +d}\, \left (B a b e x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-B \,b^{2} d x \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+B \,a^{2} e \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )-B a b d \ln \left (\frac {2 b e x +a e +b d +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}}{2 \sqrt {b e}}\right )+2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, A b -2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, B a \right )}{\sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b x +a}\, b} \]
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 \[ \int \frac {A+B\,x}{{\left (a+b\,x\right )}^{3/2}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {A + B x}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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