Optimal. Leaf size=84 \[ -\frac{4 b \sqrt{a+b x} (4 A b-5 a B)}{15 a^3 \sqrt{x}}+\frac{2 \sqrt{a+b x} (4 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A \sqrt{a+b x}}{5 a x^{5/2}} \]
[Out]
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Rubi [A] time = 0.103743, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{4 b \sqrt{a+b x} (4 A b-5 a B)}{15 a^3 \sqrt{x}}+\frac{2 \sqrt{a+b x} (4 A b-5 a B)}{15 a^2 x^{3/2}}-\frac{2 A \sqrt{a+b x}}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(7/2)*Sqrt[a + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 7.76138, size = 82, normalized size = 0.98 \[ - \frac{2 A \sqrt{a + b x}}{5 a x^{\frac{5}{2}}} + \frac{2 \sqrt{a + b x} \left (4 A b - 5 B a\right )}{15 a^{2} x^{\frac{3}{2}}} - \frac{4 b \sqrt{a + b x} \left (4 A b - 5 B a\right )}{15 a^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(7/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0610483, size = 56, normalized size = 0.67 \[ -\frac{2 \sqrt{a+b x} \left (a^2 (3 A+5 B x)-2 a b x (2 A+5 B x)+8 A b^2 x^2\right )}{15 a^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(7/2)*Sqrt[a + b*x]),x]
[Out]
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Maple [A] time = 0.006, size = 53, normalized size = 0.6 \[ -{\frac{16\,A{b}^{2}{x}^{2}-20\,B{x}^{2}ab-8\,aAbx+10\,{a}^{2}Bx+6\,A{a}^{2}}{15\,{a}^{3}}\sqrt{bx+a}{x}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(7/2)/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229253, size = 72, normalized size = 0.86 \[ -\frac{2 \,{\left (3 \, A a^{2} - 2 \,{\left (5 \, B a b - 4 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} - 4 \, A a b\right )} x\right )} \sqrt{b x + a}}{15 \, a^{3} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^(7/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**(7/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221021, size = 154, normalized size = 1.83 \[ -\frac{\sqrt{b x + a}{\left ({\left (b x + a\right )}{\left (\frac{2 \,{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )}{\left (b x + a\right )}}{a^{3} b^{9}} - \frac{5 \,{\left (5 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )}}{a^{3} b^{9}}\right )} + \frac{15 \,{\left (B a^{3} b^{4} - A a^{2} b^{5}\right )}}{a^{3} b^{9}}\right )} b}{960 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^(7/2)),x, algorithm="giac")
[Out]