Optimal. Leaf size=321 \[ \frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}-\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b} \]
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Rubi [A]
time = 0.22, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1845, 1266,
788, 649, 211, 266, 1294, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {\sqrt [4]{a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt {a} e \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b}+\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 266
Rule 631
Rule 642
Rule 649
Rule 788
Rule 1176
Rule 1179
Rule 1182
Rule 1266
Rule 1294
Rule 1845
Rubi steps
\begin {align*} \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{a+b x^4} \, dx &=\int \left (\frac {x^3 \left (c+e x^2\right )}{a+b x^4}+\frac {x^4 \left (d+f x^2\right )}{a+b x^4}\right ) \, dx\\ &=\int \frac {x^3 \left (c+e x^2\right )}{a+b x^4} \, dx+\int \frac {x^4 \left (d+f x^2\right )}{a+b x^4} \, dx\\ &=\frac {f x^3}{3 b}+\frac {1}{2} \text {Subst}\left (\int \frac {x (c+e x)}{a+b x^2} \, dx,x,x^2\right )-\frac {\int \frac {x^2 \left (3 a f-3 b d x^2\right )}{a+b x^4} \, dx}{3 b}\\ &=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}+\frac {\int \frac {-3 a b d-3 a b f x^2}{a+b x^4} \, dx}{3 b^2}+\frac {\text {Subst}\left (\int \frac {-a e+b c x}{a+b x^2} \, dx,x,x^2\right )}{2 b}\\ &=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}+\frac {1}{2} c \text {Subst}\left (\int \frac {x}{a+b x^2} \, dx,x,x^2\right )-\frac {(a e) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{2 b}-\frac {\left (\sqrt {a} \left (\sqrt {b} d-\sqrt {a} f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{2 b^2}-\frac {\left (\sqrt {a} \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{2 b^2}\\ &=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}-\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {c \log \left (a+b x^4\right )}{4 b}+\frac {\left (\sqrt [4]{a} \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}{4 \sqrt {2} b^{7/4}}+\frac {\left (\sqrt [4]{a} \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}{4 \sqrt {2} b^{7/4}}-\frac {\left (\sqrt {a} \left (\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}{4 b^2}-\frac {\left (\sqrt {a} \left (\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}{4 b^2}\\ &=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}-\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b}-\frac {\left (\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}+\frac {\left (\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}\\ &=\frac {d x}{b}+\frac {e x^2}{2 b}+\frac {f x^3}{3 b}-\frac {\sqrt {a} e \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 b^{3/2}}+\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \left (\sqrt {b} d+\sqrt {a} f\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{7/4}}+\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}-\frac {\sqrt [4]{a} \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 )}{4 \sqrt {2} b^{7/4}}+\frac {c \log \left (a+b x^4\right )}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 311, normalized size = 0.97 \begin {gather*} \frac {24 b^{3/4} d x+12 b^{3/4} e x^2+8 b^{3/4} f x^3+6 \sqrt [4]{a} \left (\sqrt {2} \sqrt {b} d+2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt {2} \sqrt {a} f\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-6 \sqrt [4]{a} \left (\sqrt {2} \sqrt {b} d-2 \sqrt [4]{a} \sqrt [4]{b} e+\sqrt {2} \sqrt {a} f\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-3 \sqrt {2} \left (-\sqrt [4]{a} \sqrt {b} d+a^{3/4} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+3 \sqrt {2} \left (-\sqrt [4]{a} \sqrt {b} d+a^{3/4} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+6 b^{3/4} c \log \left (a+b x^4\right )}{24 b^{7/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 261, normalized size = 0.81
method | result | size |
risch | \(\frac {f \,x^{3}}{3 b}+\frac {e \,x^{2}}{2 b}+\frac {d x}{b}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3} b c -\textit {\_R}^{2} a f -\textit {\_R} a e -a d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(75\) |
default | \(\frac {\frac {1}{3} f \,x^{3}+\frac {1}{2} e \,x^{2}+d x}{b}+\frac {-\frac {d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8}-\frac {a e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}-\frac {a f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {c \ln \left (b \,x^{4}+a \right )}{4}}{b}\) | \(261\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 308, normalized size = 0.96 \begin {gather*} \frac {2 \, f x^{3} + 3 \, x^{2} e + 6 \, d x}{6 \, b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c - a b d + a^{\frac {3}{2}} \sqrt {b} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} c + a b d - a^{\frac {3}{2}} \sqrt {b} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} d + \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} f - 2 \, a^{\frac {3}{2}} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} d + \sqrt {2} a^{\frac {7}{4}} b^{\frac {3}{4}} f + 2 \, a^{\frac {3}{2}} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 3.75, size = 219615, normalized size = 684.16 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 308, normalized size = 0.96 \begin {gather*} \frac {c \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} e - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \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 )}{4 \, b^{4}} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} e - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \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 )}{4 \, b^{4}} - \frac {\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 )}{8 \, b^{4}} + \frac {\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 )}{8 \, b^{4}} + \frac {2 \, b^{2} f x^{3} + 3 \, b^{2} x^{2} e + 6 \, b^{2} d x}{6 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.85, size = 838, normalized size = 2.61 \begin {gather*} \left (\sum _{k=1}^4\ln \left (\frac {a^4\,f^3+2\,a^3\,b\,c\,e\,f+a^3\,b\,d^2\,f-a^3\,b\,d\,e^2+a^2\,b^2\,c^2\,d}{b^2}+\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\,\left (\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\,\left (16\,a^2\,b^2\,d-16\,a^2\,b^2\,e\,x\right )-\frac {8\,e\,f\,a^3\,b^2+8\,c\,d\,a^2\,b^3}{b^2}+\frac {x\,\left (4\,a^3\,b\,f^2-4\,a^2\,b^2\,d^2+8\,c\,e\,a^2\,b^2\right )}{b}\right )-\frac {x\,\left (a^3\,c\,f^2-2\,a^3\,d\,e\,f+a^3\,e^3+b\,a^2\,c^2\,e-b\,a^2\,c\,d^2\right )}{b}\right )\,\mathrm {root}\left (256\,b^7\,z^4-256\,b^6\,c\,z^3+64\,a\,b^4\,d\,f\,z^2+32\,a\,b^4\,e^2\,z^2+96\,b^5\,c^2\,z^2-32\,a\,b^3\,c\,d\,f\,z-16\,a^2\,b^2\,e\,f^2\,z+16\,a\,b^3\,d^2\,e\,z-16\,a\,b^3\,c\,e^2\,z-16\,b^4\,c^3\,z-4\,a^2\,b\,d\,e^2\,f+4\,a^2\,b\,c\,e\,f^2+4\,a\,b^2\,c^2\,d\,f-4\,a\,b^2\,c\,d^2\,e+2\,a^2\,b\,d^2\,f^2+2\,a\,b^2\,c^2\,e^2+a^2\,b\,e^4+a\,b^2\,d^4+a^3\,f^4+b^3\,c^4,z,k\right )\right )+\frac {e\,x^2}{2\,b}+\frac {f\,x^3}{3\,b}+\frac {d\,x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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