Optimal. Leaf size=139 \[ -\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{3/2} (b d-a e)}+\frac {4 \sqrt {a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt {d+e x} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 45, 37} \[ -\frac {2 (A b-a B)}{b \sqrt {a+b x} (d+e x)^{3/2} (b d-a e)}+\frac {4 \sqrt {a+b x} (3 a B e-4 A b e+b B d)}{3 \sqrt {d+e x} (b d-a e)^3}+\frac {2 \sqrt {a+b x} (3 a B e-4 A b e+b B d)}{3 b (d+e x)^{3/2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{3/2} (d+e x)^{5/2}} \, dx &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{3/2}}+\frac {(b B d-4 A b e+3 a B e) \int \frac {1}{\sqrt {a+b x} (d+e x)^{5/2}} \, dx}{b (b d-a e)}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{3/2}}+\frac {2 (b B d-4 A b e+3 a B e) \sqrt {a+b x}}{3 b (b d-a e)^2 (d+e x)^{3/2}}+\frac {(2 (b B d-4 A b e+3 a B e)) \int \frac {1}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx}{3 (b d-a e)^2}\\ &=-\frac {2 (A b-a B)}{b (b d-a e) \sqrt {a+b x} (d+e x)^{3/2}}+\frac {2 (b B d-4 A b e+3 a B e) \sqrt {a+b x}}{3 b (b d-a e)^2 (d+e x)^{3/2}}+\frac {4 (b B d-4 A b e+3 a B e) \sqrt {a+b x}}{3 (b d-a e)^3 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 133, normalized size = 0.96 \[ \frac {2 \left (a^2 A e^2+a^2 B e (2 d+3 e x)-2 a A b e (3 d+2 e x)+2 a b B \left (3 d^2+5 d e x+3 e^2 x^2\right )-A b^2 \left (3 d^2+12 d e x+8 e^2 x^2\right )+b^2 B d x (3 d+2 e x)\right )}{3 \sqrt {a+b x} (d+e x)^{3/2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 5.43, size = 336, normalized size = 2.42 \[ \frac {2 \, {\left (A a^{2} e^{2} + 3 \, {\left (2 \, B a b - A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} - 3 \, A a b\right )} d e + 2 \, {\left (B b^{2} d e + {\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (3 \, B b^{2} d^{2} + 2 \, {\left (5 \, B a b - 6 \, A b^{2}\right )} d e + {\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{3 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.98, size = 584, normalized size = 4.20 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (2 \, B b^{6} d^{3} {\left | b \right |} e^{2} - B a b^{5} d^{2} {\left | b \right |} e^{3} - 5 \, A b^{6} d^{2} {\left | b \right |} e^{3} - 4 \, B a^{2} b^{4} d {\left | b \right |} e^{4} + 10 \, A a b^{5} d {\left | b \right |} e^{4} + 3 \, B a^{3} b^{3} {\left | b \right |} e^{5} - 5 \, A a^{2} b^{4} {\left | b \right |} e^{5}\right )} {\left (b x + a\right )}}{b^{7} d^{5} e - 5 \, a b^{6} d^{4} e^{2} + 10 \, a^{2} b^{5} d^{3} e^{3} - 10 \, a^{3} b^{4} d^{2} e^{4} + 5 \, a^{4} b^{3} d e^{5} - a^{5} b^{2} e^{6}} + \frac {3 \, {\left (B b^{7} d^{4} {\left | b \right |} e - 2 \, B a b^{6} d^{3} {\left | b \right |} e^{2} - 2 \, A b^{7} d^{3} {\left | b \right |} e^{2} + 6 \, A a b^{6} d^{2} {\left | b \right |} e^{3} + 2 \, B a^{3} b^{4} d {\left | b \right |} e^{4} - 6 \, A a^{2} b^{5} d {\left | b \right |} e^{4} - B a^{4} b^{3} {\left | b \right |} e^{5} + 2 \, A a^{3} b^{4} {\left | b \right |} e^{5}\right )}}{b^{7} d^{5} e - 5 \, a b^{6} d^{4} e^{2} + 10 \, a^{2} b^{5} d^{3} e^{3} - 10 \, a^{3} b^{4} d^{2} e^{4} + 5 \, a^{4} b^{3} d e^{5} - a^{5} b^{2} e^{6}}\right )}}{3 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (B^{2} a^{2} b^{5} e - 2 \, A B a b^{6} e + A^{2} b^{7} e\right )}}{{\left (B a b^{\frac {9}{2}} d e^{\frac {1}{2}} - A b^{\frac {11}{2}} d e^{\frac {1}{2}} - B a^{2} b^{\frac {7}{2}} e^{\frac {3}{2}} + A a b^{\frac {9}{2}} e^{\frac {3}{2}} - {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} B a b^{\frac {5}{2}} e^{\frac {1}{2}} + {\left (\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}\right )}^{2} A b^{\frac {7}{2}} e^{\frac {1}{2}}\right )} {\left (b^{2} d^{2} {\left | b \right |} - 2 \, a b d {\left | b \right |} e + a^{2} {\left | b \right |} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 176, normalized size = 1.27 \[ -\frac {2 \left (-8 A \,b^{2} e^{2} x^{2}+6 B a b \,e^{2} x^{2}+2 B \,b^{2} d e \,x^{2}-4 A a b \,e^{2} x -12 A \,b^{2} d e x +3 B \,a^{2} e^{2} x +10 B a b d e x +3 B \,b^{2} d^{2} x +A \,a^{2} e^{2}-6 A a b d e -3 A \,b^{2} d^{2}+2 B \,a^{2} d e +6 B a b \,d^{2}\right )}{3 \sqrt {b x +a}\, \left (e x +d \right )^{\frac {3}{2}} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )} \]
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 = 2.20, size = 182, normalized size = 1.31 \[ -\frac {\sqrt {d+e\,x}\,\left (\frac {4\,B\,a^2\,d\,e+2\,A\,a^2\,e^2+12\,B\,a\,b\,d^2-12\,A\,a\,b\,d\,e-6\,A\,b^2\,d^2}{3\,e^2\,{\left (a\,e-b\,d\right )}^3}+\frac {2\,x\,\left (a\,e+3\,b\,d\right )\,\left (3\,B\,a\,e-4\,A\,b\,e+B\,b\,d\right )}{3\,e^2\,{\left (a\,e-b\,d\right )}^3}+\frac {4\,b\,x^2\,\left (3\,B\,a\,e-4\,A\,b\,e+B\,b\,d\right )}{3\,e\,{\left (a\,e-b\,d\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {d^2\,\sqrt {a+b\,x}}{e^2}+\frac {2\,d\,x\,\sqrt {a+b\,x}}{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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