Optimal. Leaf size=89 \[ -\frac{2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b B \sqrt{a+b x}}{\sqrt{x}} \]
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Rubi [A] time = 0.0901926, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 A (a+b x)^{5/2}}{5 a x^{5/2}}+2 b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 B (a+b x)^{3/2}}{3 x^{3/2}}-\frac{2 b B \sqrt{a+b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/x^(7/2),x]
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Rubi in Sympy [A] time = 9.42584, size = 85, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{5 a x^{\frac{5}{2}}} + 2 B b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )} - \frac{2 B b \sqrt{a + b x}}{\sqrt{x}} - \frac{2 B \left (a + b x\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.146091, size = 89, normalized size = 1. \[ 2 b^{3/2} B \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{2 \sqrt{a+b x} \left (a^2 (3 A+5 B x)+2 a b x (3 A+10 B x)+3 A b^2 x^2\right )}{15 a x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(7/2),x]
[Out]
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Maple [B] time = 0.02, size = 143, normalized size = 1.6 \[ -{\frac{1}{15\,a}\sqrt{bx+a} \left ( -15\,B{b}^{3/2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{3}+6\,A{x}^{2}{b}^{2}\sqrt{x \left ( bx+a \right ) }+40\,B{x}^{2}ab\sqrt{x \left ( bx+a \right ) }+12\,Axab\sqrt{x \left ( bx+a \right ) }+10\,Bx{a}^{2}\sqrt{x \left ( bx+a \right ) }+6\,A{a}^{2}\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/x^(7/2),x)
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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)^(3/2)/x^(7/2),x, algorithm="maxima")
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Fricas [A] time = 0.247758, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a b^{\frac{3}{2}} x^{3} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (3 \, A a^{2} +{\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{15 \, a x^{3}}, \frac{2 \,{\left (15 \, B a \sqrt{-b} b x^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) -{\left (3 \, A a^{2} +{\left (20 \, B a b + 3 \, A b^{2}\right )} x^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{15 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/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)**(3/2)*(B*x+A)/x**(7/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x^(7/2),x, algorithm="giac")
[Out]