Optimal. Leaf size=89 \[ -\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}+\frac {2 (b B d-2 A b e+a B e) \sqrt {d+e x}}{e (b d-a e)^2 \sqrt {a+b x}} \]
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Rubi [A]
time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {79, 37}
\begin {gather*} \frac {2 \sqrt {d+e x} (a B e-2 A b e+b B d)}{e \sqrt {a+b x} (b d-a e)^2}-\frac {2 (B d-A e)}{e \sqrt {a+b x} \sqrt {d+e x} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{3/2}} \, dx &=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}-\frac {(b B d-2 A b e+a B e) \int \frac {1}{(a+b x)^{3/2} \sqrt {d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac {2 (B d-A e)}{e (b d-a e) \sqrt {a+b x} \sqrt {d+e x}}+\frac {2 (b B d-2 A b e+a B e) \sqrt {d+e x}}{e (b d-a e)^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 61, normalized size = 0.69 \begin {gather*} \frac {2 B (2 a d+b d x+a e x)-2 A (a e+b (d+2 e x))}{(b d-a e)^2 \sqrt {a+b x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 59, normalized size = 0.66
method | result | size |
default | \(-\frac {2 \left (2 A b e x -B a e x -B b d x +A a e +A b d -2 B a d \right )}{\left (a e -b d \right )^{2} \sqrt {b x +a}\, \sqrt {e x +d}}\) | \(59\) |
gosper | \(-\frac {2 \left (2 A b e x -B a e x -B b d x +A a e +A b d -2 B a d \right )}{\sqrt {b x +a}\, \sqrt {e x +d}\, \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )}\) | \(72\) |
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 [A]
time = 1.12, size = 150, normalized size = 1.69 \begin {gather*} \frac {2 \, {\left (B b d x + {\left (2 \, B a - A b\right )} d - {\left (A a - {\left (B a - 2 \, A b\right )} x\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d}}{b^{3} d^{3} x + a b^{2} d^{3} + {\left (a^{2} b x^{2} + a^{3} x\right )} e^{3} - {\left (2 \, a b^{2} d x^{2} + a^{2} b d x - a^{3} d\right )} e^{2} + {\left (b^{3} d^{2} x^{2} - a b^{2} d^{2} x - 2 \, a^{2} b d^{2}\right )} e} \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}{\left (a + b x\right )^{\frac {3}{2}} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs.
\(2 (86) = 172\).
time = 0.55, size = 264, normalized size = 2.97 \begin {gather*} \frac {2 \, {\left (B b^{2} d - A b^{2} e\right )} \sqrt {b x + a}}{{\left (b^{2} d^{2} {\left | b \right |} - 2 \, a b d {\left | b \right |} e + a^{2} {\left | b \right |} e^{2}\right )} \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} + \frac {4 \, {\left (B^{2} a^{2} b^{3} e - 2 \, A B a b^{4} e + A^{2} b^{5} e\right )}}{{\left (B a b^{\frac {7}{2}} d e^{\frac {1}{2}} - A b^{\frac {9}{2}} d e^{\frac {1}{2}} - B a^{2} b^{\frac {5}{2}} e^{\frac {3}{2}} + A a b^{\frac {7}{2}} e^{\frac {3}{2}} - {\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} B a b^{\frac {3}{2}} e^{\frac {1}{2}} + {\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} A b^{\frac {5}{2}} e^{\frac {1}{2}}\right )} {\left (b d {\left | b \right |} - a {\left | b \right |} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.00, size = 96, normalized size = 1.08 \begin {gather*} -\frac {\left (\frac {2\,A\,a\,e+2\,A\,b\,d-4\,B\,a\,d}{e\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\left (2\,B\,a\,e-4\,A\,b\,e+2\,B\,b\,d\right )}{e\,{\left (a\,e-b\,d\right )}^2}\right )\,\sqrt {d+e\,x}}{x\,\sqrt {a+b\,x}+\frac {d\,\sqrt {a+b\,x}}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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