Optimal. Leaf size=140 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{a+b x} (7 A b-4 a B)}{4 a^4 x}-\frac{5 (7 A b-4 a B)}{6 a^3 x \sqrt{a+b x}}-\frac{7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac{A}{2 a x^2 (a+b x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.183355, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{5 b (7 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{5 \sqrt{a+b x} (7 A b-4 a B)}{4 a^4 x}-\frac{5 (7 A b-4 a B)}{6 a^3 x \sqrt{a+b x}}-\frac{7 A b-4 a B}{6 a^2 x (a+b x)^{3/2}}-\frac{A}{2 a x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a + b*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 16.9104, size = 129, normalized size = 0.92 \[ - \frac{A}{2 a x^{2} \left (a + b x\right )^{\frac{3}{2}}} - \frac{7 A b - 4 B a}{6 a^{2} x \left (a + b x\right )^{\frac{3}{2}}} - \frac{5 \left (7 A b - 4 B a\right )}{6 a^{3} x \sqrt{a + b x}} + \frac{5 \sqrt{a + b x} \left (7 A b - 4 B a\right )}{4 a^{4} x} - \frac{5 b \left (7 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**3/(b*x+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.171861, size = 107, normalized size = 0.76 \[ \frac{5 b (4 a B-7 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{-6 a^3 (A+2 B x)+a^2 b x (21 A-80 B x)+20 a b^2 x^2 (7 A-3 B x)+105 A b^3 x^3}{12 a^4 x^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a + b*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.024, size = 122, normalized size = 0.9 \[ 2\,b \left ( -{\frac{-3\,Ab+2\,Ba}{{a}^{4}\sqrt{bx+a}}}-1/3\,{\frac{-Ab+Ba}{{a}^{3} \left ( bx+a \right ) ^{3/2}}}+{\frac{1}{{a}^{4}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( \left ({\frac{11\,Ab}{8}}-1/2\,Ba \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{13\,Aab}{8}}+1/2\,B{a}^{2} \right ) \sqrt{bx+a} \right ) }-5/8\,{\frac{7\,Ab-4\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(b*x+a)^(5/2),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)/((b*x + a)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235656, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left ({\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (6 \, A a^{3} + 15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{a}}{24 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{a}}, -\frac{15 \,{\left ({\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (6 \, A a^{3} + 15 \,{\left (4 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 20 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 3 \,{\left (4 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{-a}}{12 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{b x + a} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(5/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 58.8049, size = 1287, normalized size = 9.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**3/(b*x+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217608, size = 201, normalized size = 1.44 \[ -\frac{5 \,{\left (4 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{4}} - \frac{2 \,{\left (6 \,{\left (b x + a\right )} B a b + B a^{2} b - 9 \,{\left (b x + a\right )} A b^{2} - A a b^{2}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}} - \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x + a} B a^{2} b - 11 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{2} + 13 \, \sqrt{b x + a} A a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b*x + a)^(5/2)*x^3),x, algorithm="giac")
[Out]