Optimal. Leaf size=340 \[ -\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]
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Rubi [A] time = 0.328179, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {1823, 1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{a} f+\sqrt{b} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1823
Rule 1855
Rule 1876
Rule 275
Rule 205
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{\int \frac{d+2 e x+3 f x^2}{\left (a+b x^4\right )^2} \, dx}{8 b}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac{\int \frac{-3 d-4 e x-3 f x^2}{a+b x^4} \, dx}{32 a b}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac{\int \left (-\frac{4 e x}{a+b x^4}+\frac{-3 d-3 f x^2}{a+b x^4}\right ) \, dx}{32 a b}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}-\frac{\int \frac{-3 d-3 f x^2}{a+b x^4} \, dx}{32 a b}+\frac{e \int \frac{x}{a+b x^4} \, dx}{8 a b}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac{e \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^2\right )}{16 a b}+\frac{\left (3 \left (\frac{\sqrt{b} d}{\sqrt{a}}-f\right )\right ) \int \frac{\sqrt{a} \sqrt{b}-b x^2}{a+b x^4} \, dx}{64 a b^2}+\frac{\left (3 \left (\frac{\sqrt{b} d}{\sqrt{a}}+f\right )\right ) \int \frac{\sqrt{a} \sqrt{b}+b x^2}{a+b x^4} \, dx}{64 a b^2}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}+\frac{\left (3 \left (\frac{\sqrt{b} d}{\sqrt{a}}+f\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}+\frac{\left (3 \left (\frac{\sqrt{b} d}{\sqrt{a}}+f\right )\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a b^2}-\frac{\left (3 \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \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}{128 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\left (3 \left (\sqrt{b} d-\sqrt{a} f\right )\right ) \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}{128 \sqrt{2} a^{7/4} b^{7/4}}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{\left (3 \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}-\frac{\left (3 \left (\sqrt{b} d+\sqrt{a} f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}\\ &=-\frac{c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}+\frac{x \left (d+2 e x+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{16 a^{3/2} b^{3/2}}-\frac{3 \left (\sqrt{b} d+\sqrt{a} f\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d+\sqrt{a} f\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} b^{7/4}}-\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}+\frac{3 \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{b} x^2\right )}{128 \sqrt{2} a^{7/4} b^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.349547, size = 329, normalized size = 0.97 \[ \frac{-\frac{2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right )}{a^{7/4}}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt{2} \sqrt{a} f+3 \sqrt{2} \sqrt{b} d\right )}{a^{7/4}}+\frac{3 \sqrt{2} \left (\sqrt{a} f-\sqrt{b} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}+\frac{3 \sqrt{2} \left (\sqrt{b} d-\sqrt{a} f\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt{a}+\sqrt{b} x^2\right )}{a^{7/4}}-\frac{32 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^2}+\frac{8 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )}}{256 b^{7/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 373, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ( b{x}^{4}+a \right ) ^{2}} \left ({\frac{3\,f{x}^{7}}{32\,a}}+{\frac{e{x}^{6}}{16\,a}}+{\frac{d{x}^{5}}{32\,a}}-{\frac{f{x}^{3}}{32\,b}}-{\frac{e{x}^{2}}{16\,b}}-{\frac{3\,dx}{32\,b}}-{\frac{c}{8\,b}} \right ) }+{\frac{3\,d\sqrt{2}}{256\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,d\sqrt{2}}{128\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,d\sqrt{2}}{128\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{e}{16\,ab}\arctan \left ({x}^{2}\sqrt{{\frac{b}{a}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,f\sqrt{2}}{256\,{b}^{2}a}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{b}}}x\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{128\,{b}^{2}a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,f\sqrt{2}}{128\,{b}^{2}a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \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 [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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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 = 1.09603, size = 456, normalized size = 1.34 \begin{align*} \frac{3 \, b f x^{7} + 2 \, b x^{6} e + b d x^{5} - a f x^{3} - 2 \, a x^{2} e - 3 \, a d x - 4 \, a c}{32 \,{\left (b x^{4} + a\right )}^{2} a b} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (4 \, \sqrt{2} \sqrt{a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} + \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{2} b^{4}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac{3}{4}} f\right )} \log \left (x^{2} - \sqrt{2} x \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{256 \, a^{2} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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