Optimal. Leaf size=340 \[ -\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{2 d^2 \sqrt{x} (3 b c-2 a d)}{b^3}+\frac{\sqrt{x} (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{2 d^3 x^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.388311, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {466, 390, 385, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{2 d^2 \sqrt{x} (3 b c-2 a d)}{b^3}+\frac{\sqrt{x} (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{2 d^3 x^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 466
Rule 390
Rule 385
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^3}{\sqrt{x} \left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{d^2 (3 b c-2 a d)}{b^3}+\frac{d^3 x^4}{b^2}+\frac{(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{b^3 \left (a+b x^4\right )^2}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{2 \operatorname{Subst}\left (\int \frac{(b c-a d)^2 (b c+2 a d)+3 b d (b c-a d)^2 x^4}{\left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{(b c-a d)^3 \sqrt{x}}{2 a b^3 \left (a+b x^2\right )}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a b^3}\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{(b c-a d)^3 \sqrt{x}}{2 a b^3 \left (a+b x^2\right )}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} b^3}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} b^3}\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{(b c-a d)^3 \sqrt{x}}{2 a b^3 \left (a+b x^2\right )}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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 )}{8 a^{3/2} b^{7/2}}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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 )}{8 a^{3/2} b^{7/2}}-\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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 )}{8 \sqrt{2} a^{7/4} b^{13/4}}-\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \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 )}{8 \sqrt{2} a^{7/4} b^{13/4}}\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{(b c-a d)^3 \sqrt{x}}{2 a b^3 \left (a+b x^2\right )}-\frac{3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}-\frac{\left (3 (b c-a d)^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}\\ &=\frac{2 d^2 (3 b c-2 a d) \sqrt{x}}{b^3}+\frac{2 d^3 x^{5/2}}{5 b^2}+\frac{(b c-a d)^3 \sqrt{x}}{2 a b^3 \left (a+b x^2\right )}-\frac{3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (b c+3 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{13/4}}-\frac{3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}+\frac{3 (b c-a d)^2 (b c+3 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{13/4}}\\ \end{align*}
Mathematica [C] time = 2.01997, size = 358, normalized size = 1.05 \[ \frac{a \left (45 a^2 \left (85683 c^2 d x^2+28561 c^3+85683 c d^2 x^4+25105 d^3 x^6\right )+18 a b x^2 \left (104781 c^2 d x^2+34927 c^3+119181 c d^2 x^4+36655 d^3 x^6\right )+b^2 x^4 \left (98259 c^2 d x^2+50033 c^3+98259 c d^2 x^4+32753 d^3 x^6\right )\right )-45 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right ) \left (9 a^2 b x^2 \left (6561 c^2 d x^2+2187 c^3+7201 c d^2 x^4+2187 d^3 x^6\right )+a^3 \left (85683 c^2 d x^2+28561 c^3+85683 c d^2 x^4+25105 d^3 x^6\right )+3 a b^2 x^4 \left (1491 c^2 d x^2+625 c^3+1875 c d^2 x^4+625 d^3 x^6\right )+b^3 x^6 \left (3 c^2 d x^2-1151 c^3+3 c d^2 x^4+d^3 x^6\right )\right )}{34560 a^2 b^3 x^{11/2}}-\frac{128 b x^{5/2} \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{\frac{5}{4},2,2,2,2\right \},\left \{1,1,1,\frac{21}{4}\right \},-\frac{b x^2}{a}\right )}{9945 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.016, size = 697, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 1.37465, size = 4316, normalized size = 12.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.2229, size = 690, normalized size = 2.03 \begin{align*} \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\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 )}{8 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\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 )}{8 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} + \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d - 5 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac{b^{3} c^{3} \sqrt{x} - 3 \, a b^{2} c^{2} d \sqrt{x} + 3 \, a^{2} b c d^{2} \sqrt{x} - a^{3} d^{3} \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a b^{3}} + \frac{2 \,{\left (b^{8} d^{3} x^{\frac{5}{2}} + 15 \, b^{8} c d^{2} \sqrt{x} - 10 \, a b^{7} d^{3} \sqrt{x}\right )}}{5 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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