Optimal. Leaf size=94 \[ \frac {2 \sqrt {a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt {d+e x} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.05, antiderivative size = 94, 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} \[ \frac {2 \sqrt {a+b x} (-3 a B e+2 A b e+b B d)}{3 e \sqrt {d+e x} (b d-a e)^2}-\frac {2 \sqrt {a+b x} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e) \sqrt {a+b x}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {(b B d+2 A b e-3 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac {2 (B d-A e) \sqrt {a+b x}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {2 (b B d+2 A b e-3 a B e) \sqrt {a+b x}}{3 e (b d-a e)^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.69 \[ \frac {2 \sqrt {a+b x} (A (-a e+3 b d+2 b e x)+B (-2 a d-3 a e x+b d x))}{3 (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.63, size = 140, normalized size = 1.49 \[ -\frac {2 \, {\left (A a e + {\left (2 \, B a - 3 \, A b\right )} d - {\left (B b d - {\left (3 \, B a - 2 \, A b\right )} e\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 179, normalized size = 1.90 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (B b^{4} d {\left | b \right |} e - 3 \, B a b^{3} {\left | b \right |} e^{2} + 2 \, A b^{4} {\left | b \right |} e^{2}\right )} {\left (b x + a\right )}}{b^{4} d^{2} e - 2 \, a b^{3} d e^{2} + a^{2} b^{2} e^{3}} - \frac {3 \, {\left (B a b^{4} d {\left | b \right |} e - A b^{5} d {\left | b \right |} e - B a^{2} b^{3} {\left | b \right |} e^{2} + A a b^{4} {\left | b \right |} e^{2}\right )}}{b^{4} d^{2} e - 2 \, a b^{3} d e^{2} + a^{2} b^{2} e^{3}}\right )}}{3 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.78 \[ -\frac {2 \sqrt {b x +a}\, \left (-2 A b e x +3 B a e x -B b d x +A a e -3 A b d +2 B a d \right )}{3 \left (e x +d \right )^{\frac {3}{2}} \left (a^{2} e^{2}-2 b e a d +b^{2} d^{2}\right )} \]
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 = 2.04, size = 169, normalized size = 1.80 \[ \frac {\sqrt {d+e\,x}\,\left (\frac {x\,\left (6\,A\,b^2\,d-6\,B\,a^2\,e+2\,A\,a\,b\,e-2\,B\,a\,b\,d\right )}{3\,e^2\,{\left (a\,e-b\,d\right )}^2}-\frac {2\,A\,a^2\,e+4\,B\,a^2\,d-6\,A\,a\,b\,d}{3\,e^2\,{\left (a\,e-b\,d\right )}^2}+\frac {x^2\,\left (4\,A\,b^2\,e+2\,B\,b^2\,d-6\,B\,a\,b\,e\right )}{3\,e^2\,{\left (a\,e-b\,d\right )}^2}\right )}{x^2\,\sqrt {a+b\,x}+\frac {d^2\,\sqrt {a+b\,x}}{e^2}+\frac {2\,d\,x\,\sqrt {a+b\,x}}{e}} \]
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 \[ \int \frac {A + B x}{\sqrt {a + b x} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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