Optimal. Leaf size=315 \[ \frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac{\sqrt{x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.589067, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ \frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{7/4} c^{7/4}}-\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}+\frac{3 \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac{\sqrt{x} (A+3 B x)}{16 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x))/(a + c*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 112.037, size = 296, normalized size = 0.94 \[ - \frac{\sqrt{x} \left (2 A + 2 B x\right )}{8 c \left (a + c x^{2}\right )^{2}} + \frac{\sqrt{x} \left (A + 3 B x\right )}{16 a c \left (a + c x^{2}\right )} - \frac{3 \sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{7}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{128 a^{\frac{7}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} c^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)/(c*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.580643, size = 296, normalized size = 0.94 \[ \frac{\frac{3 \sqrt{2} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{7/4}}-\frac{3 \sqrt{2} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{a^{7/4}}-\frac{6 \sqrt{2} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{32 c^{3/4} \sqrt{x} (A+B x)}{\left (a+c x^2\right )^2}+\frac{8 c^{3/4} \sqrt{x} (A+3 B x)}{a \left (a+c x^2\right )}}{128 c^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x))/(a + c*x^2)^3,x]
[Out]
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Maple [A] time = 0.025, size = 334, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,B{x}^{7/2}}{32\,a}}+1/32\,{\frac{A{x}^{5/2}}{a}}-1/32\,{\frac{B{x}^{3/2}}{c}}-{\frac{3\,A\sqrt{x}}{32\,c}} \right ) }+{\frac{3\,A\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,A\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,A\sqrt{2}}{64\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,B\sqrt{2}}{128\,a{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{64\,a{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{64\,a{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)/(c*x^2+a)^3,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)*x^(3/2)/(c*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.318202, size = 1326, normalized size = 4.21 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(c*x^2 + a)^3,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(x**(3/2)*(B*x+A)/(c*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.284131, size = 390, normalized size = 1.24 \[ \frac{3 \, B c x^{\frac{7}{2}} + A c x^{\frac{5}{2}} - B a x^{\frac{3}{2}} - 3 \, A a \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} a c} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{2} c^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(c*x^2 + a)^3,x, algorithm="giac")
[Out]