Optimal. Leaf size=85 \[ -\frac {2 (B d-A e) \sqrt {a+b x}}{e (b d-a e) \sqrt {d+e x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}} \]
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Rubi [A]
time = 0.03, 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 = {79, 65, 223,
212} \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}-\frac {2 \sqrt {a+b x} (B d-A e)}{e \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e) \sqrt {a+b x}}{e (b d-a e) \sqrt {d+e x}}+\frac {B \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{e}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{e (b d-a e) \sqrt {d+e x}}+\frac {(2 B) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b e}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{e (b d-a e) \sqrt {d+e x}}+\frac {(2 B) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{b e}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{e (b d-a e) \sqrt {d+e x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 85, normalized size = 1.00 \begin {gather*} -\frac {2 (-B d+A e) \sqrt {a+b x}}{e (-b d+a e) \sqrt {d+e x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(69)=138\).
time = 0.11, size = 278, normalized size = 3.27
method | result | size |
default | \(\frac {\left (B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,e^{2} x -B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b d e x +B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a d e -B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b \,d^{2}-2 A e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+2 B d \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right ) \sqrt {b x +a}}{\sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, e \sqrt {e x +d}}\) | \(278\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (70) = 140\).
time = 0.74, size = 356, normalized size = 4.19 \begin {gather*} \left [\frac {{\left (B b d^{2} - B a x e^{2} + {\left (B b d x - B a d\right )} e\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (B b d e - A b e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{2 \, {\left (b^{2} d^{2} e^{2} - a b x e^{4} + {\left (b^{2} d x - a b d\right )} e^{3}\right )}}, -\frac {{\left (B b d^{2} - B a x e^{2} + {\left (B b d x - B a d\right )} e\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (B b d e - A b e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}}{b^{2} d^{2} e^{2} - a b x e^{4} + {\left (b^{2} d x - a b d\right )} e^{3}}\right ] \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 {a + b x} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 120, normalized size = 1.41 \begin {gather*} -\frac {2 \, B {\left | b \right |} e^{\left (-\frac {3}{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 |}\right )}{b^{\frac {3}{2}}} - \frac {2 \, {\left (B b^{2} d {\left | b \right |} - A b^{2} {\left | b \right |} e\right )} \sqrt {b x + a}}{{\left (b^{3} d e - a b^{2} e^{2}\right )} \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} \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 {a+b\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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