Optimal. Leaf size=298 \[ -\frac{(5 a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{\sqrt{x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.211935, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {457, 288, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(5 a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{\sqrt{x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 457
Rule 288
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}+\frac{\left (\frac{3 A b}{2}+\frac{5 a B}{2}\right ) \int \frac{x^{3/2}}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}+\frac{(3 A b+5 a B) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{32 a b^2}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{16 a b^2}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^{3/2} b^2}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^{3/2} b^2}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^{3/2} b^{5/2}}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^{3/2} b^{5/2}}-\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}-\frac{(3 A b+5 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(3 A b+5 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(3 A b+5 a B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}\\ &=\frac{(A b-a B) x^{5/2}}{4 a b \left (a+b x^2\right )^2}-\frac{(3 A b+5 a B) \sqrt{x}}{16 a b^2 \left (a+b x^2\right )}-\frac{(3 A b+5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(3 A b+5 a B) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(3 A b+5 a B) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(3 A b+5 a B) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.442023, size = 389, normalized size = 1.31 \[ \frac{\frac{8 A b^{5/4} \sqrt{x}}{a^2+a b x^2}-\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{3 \sqrt{2} A b \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}+\frac{3 \sqrt{2} A b \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}-\frac{5 \sqrt{2} B \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{5 \sqrt{2} B \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}-\frac{32 A b^{5/4} \sqrt{x}}{\left (a+b x^2\right )^2}-\frac{72 \sqrt [4]{b} B \sqrt{x}}{a+b x^2}+\frac{32 a \sqrt [4]{b} B \sqrt{x}}{\left (a+b x^2\right )^2}}{128 b^{9/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 334, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( Ab-9\,Ba \right ){x}^{5/2}}{ab}}-1/32\,{\frac{ \left ( 3\,Ab+5\,Ba \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{3\,\sqrt{2}A}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}A}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}A}{128\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}B}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}B}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}B}{128\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.0764, size = 1817, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16297, size = 402, normalized size = 1.35 \begin{align*} \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac{9 \, B a b x^{\frac{5}{2}} - A b^{2} x^{\frac{5}{2}} + 5 \, B a^{2} \sqrt{x} + 3 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]