Optimal. Leaf size=305 \[ -\frac{2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}+\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{2 c^3}{11 a x^{11/2}} \]
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Rubi [A] time = 0.267611, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {466, 461, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 c \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{3 a^3 x^{3/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}+\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{2 c^3}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 466
Rule 461
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}{x^{13/2} \left (a+b x^2\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (c+d x^4\right )^3}{x^{12} \left (a+b x^4\right )} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{c^3}{a x^{12}}+\frac{c^2 (-b c+3 a d)}{a^2 x^8}+\frac{c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x^4}+\frac{(-b c+a d)^3}{a^3 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 c^3}{11 a x^{11/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac{\left (2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=-\frac{2 c^3}{11 a x^{11/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^{7/2}}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^{7/2}}\\ &=-\frac{2 c^3}{11 a x^{11/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}-\frac{(b c-a d)^3 \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 )}{2 a^{7/2} \sqrt{b}}-\frac{(b c-a d)^3 \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 )}{2 a^{7/2} \sqrt{b}}+\frac{(b c-a d)^3 \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 )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(b c-a d)^3 \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 )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac{2 c^3}{11 a x^{11/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac{(b c-a d)^3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}\\ &=-\frac{2 c^3}{11 a x^{11/2}}+\frac{2 c^2 (b c-3 a d)}{7 a^2 x^{7/2}}-\frac{2 c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{3 a^3 x^{3/2}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{(b c-a d)^3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{(b c-a d)^3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{15/4} \sqrt [4]{b}}\\ \end{align*}
Mathematica [C] time = 0.358459, size = 102, normalized size = 0.33 \[ -\frac{2 \left (a c \left (3 a^2 \left (7 c^2+33 c d x^2+77 d^2 x^4\right )-33 a b c x^2 \left (c+7 d x^2\right )+77 b^2 c^2 x^4\right )+231 x^6 (b c-a d)^3 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right )\right )}{231 a^4 x^{11/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 659, normalized size = 2.2 \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 = 1.3615, size = 4089, normalized size = 13.41 \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 [B] time = 1.16721, size = 652, normalized size = 2.14 \begin{align*} -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{2 \, a^{4} b} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{2 \, a^{4} b} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{4 \, a^{4} b} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \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 )}{4 \, a^{4} b} - \frac{2 \,{\left (77 \, b^{2} c^{3} x^{4} - 231 \, a b c^{2} d x^{4} + 231 \, a^{2} c d^{2} x^{4} - 33 \, a b c^{3} x^{2} + 99 \, a^{2} c^{2} d x^{2} + 21 \, a^{2} c^{3}\right )}}{231 \, a^{3} x^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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