Optimal. Leaf size=85 \[ -\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b x}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}} \]
[Out]
________________________________________________________________________________________
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 )}{b^{3/2} \sqrt {e}}-\frac {2 \sqrt {d+e x} (A b-a B)}{b \sqrt {a+b x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 79
Rule 212
Rule 223
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) \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^2}\\ &=-\frac {2 (A b-a B) \sqrt {d+e x}}{b (b d-a e) \sqrt {a+b 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^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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 85, normalized size = 1.00 \begin {gather*} -\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 {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{b^{3/2} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(69)=138\).
time = 0.09, size = 278, normalized size = 3.27
method | result | size |
default | \(\frac {\sqrt {e x +d}\, \left (B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b e x -B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d x +B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e -B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d +2 A b \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-2 B a \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{\sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b \sqrt {b x +a}}\) | \(278\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 177 vs.
\(2 (70) = 140\).
time = 1.08, size = 363, normalized size = 4.27 \begin {gather*} \left [-\frac {{\left (B b^{2} d x + B a b d - {\left (B a b x + B a^{2}\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 a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {x e + d} e}{2 \, {\left ({\left (a b^{3} x + a^{2} b^{2}\right )} e^{2} - {\left (b^{4} d x + a b^{3} d\right )} e\right )}}, -\frac {2 \, {\left (B a b - A b^{2}\right )} \sqrt {b x + a} \sqrt {x e + d} e - {\left (B b^{2} d x + B a b d - {\left (B a b x + B a^{2}\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 )}{{\left (a b^{3} x + a^{2} b^{2}\right )} e^{2} - {\left (b^{4} d x + a b^{3} d\right )} e}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.60, size = 135, normalized size = 1.59 \begin {gather*} -\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 |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,x}{{\left (a+b\,x\right )}^{3/2}\,\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________