Optimal. Leaf size=95 \[ -\frac {2 \sqrt {d+e x} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}-\frac {2 \sqrt {d+e x} (-a B e-2 A b e+3 b B d)}{3 b \sqrt {a+b x} (b d-a e)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 95, 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 = {78, 37} \[ -\frac {2 \sqrt {d+e x} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}-\frac {2 \sqrt {d+e x} (-a B e-2 A b e+3 b B d)}{3 b \sqrt {a+b x} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2} \sqrt {d+e x}} \, dx &=-\frac {2 (A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {(3 b B d-2 A b e-a B e) \int \frac {1}{(a+b x)^{3/2} \sqrt {d+e x}} \, dx}{3 b (b d-a e)}\\ &=-\frac {2 (A b-a B) \sqrt {d+e x}}{3 b (b d-a e) (a+b x)^{3/2}}-\frac {2 (3 b B d-2 A b e-a B e) \sqrt {d+e x}}{3 b (b d-a e)^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.67 \[ -\frac {2 \sqrt {d+e x} (-3 a A e+B (2 a d-a e x+3 b d x)+A b (d-2 e x))}{3 (a+b x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.05, size = 141, normalized size = 1.48 \[ \frac {2 \, {\left (3 \, A a e - {\left (2 \, B a + A b\right )} d - {\left (3 \, B b d - {\left (B a + 2 \, A b\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (a^{2} b^{2} d^{2} - 2 \, a^{3} b d e + a^{4} e^{2} + {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.41, size = 261, normalized size = 2.75 \[ -\frac {4 \, {\left (3 \, B b^{\frac {9}{2}} d^{2} e^{\frac {1}{2}} - 4 \, B a b^{\frac {7}{2}} d e^{\frac {3}{2}} - 2 \, A b^{\frac {9}{2}} d e^{\frac {3}{2}} - 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B b^{\frac {5}{2}} d e^{\frac {1}{2}} + B a^{2} b^{\frac {5}{2}} e^{\frac {5}{2}} + 2 \, A a b^{\frac {7}{2}} e^{\frac {5}{2}} + 6 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac {5}{2}} e^{\frac {3}{2}} + 3 \, {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{4} B \sqrt {b} e^{\frac {1}{2}}\right )}}{3 \, {\left (b^{2} d - a b e - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2}\right )}^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.77 \[ \frac {2 \sqrt {e x +d}\, \left (2 A b e x +B a e x -3 B b d x +3 A a e -A b d -2 B a d \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )} \]
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 [B] time = 2.02, size = 97, normalized size = 1.02 \[ -\frac {\left (\frac {2\,A\,b\,d-6\,A\,a\,e+4\,B\,a\,d}{3\,b\,{\left (a\,e-b\,d\right )}^2}-\frac {x\,\left (4\,A\,b\,e+2\,B\,a\,e-6\,B\,b\,d\right )}{3\,b\,{\left (a\,e-b\,d\right )}^2}\right )\,\sqrt {d+e\,x}}{x\,\sqrt {a+b\,x}+\frac {a\,\sqrt {a+b\,x}}{b}} \]
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 \[ \int \frac {A + B x}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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