Optimal. Leaf size=97 \[ \frac {e x^3}{3 c}+\frac {\left (b c d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1488, 787, 648,
632, 212, 642} \begin {gather*} \frac {\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac {e x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 787
Rule 1488
Rubi steps
\begin {align*} \int \frac {x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x (d+e x)}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac {e x^3}{3 c}+\frac {\text {Subst}\left (\int \frac {-a e+(c d-b e) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c}\\ &=\frac {e x^3}{3 c}+\frac {(c d-b e) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}-\frac {\left (b c d-b^2 e+2 a c e\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}\\ &=\frac {e x^3}{3 c}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac {\left (b c d-b^2 e+2 a c e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^2}\\ &=\frac {e x^3}{3 c}+\frac {\left (b c d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^3}{\sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c}}+\frac {(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.96 \begin {gather*} \frac {2 c e x^3+\frac {2 \left (-b c d+b^2 e-2 a c e\right ) \tan ^{-1}\left (\frac {b+2 c x^3}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 98, normalized size = 1.01
method | result | size |
default | \(\frac {e \,x^{3}}{3 c}+\frac {\frac {\left (-e b +c d \right ) \ln \left (c \,x^{6}+b \,x^{3}+a \right )}{2 c}+\frac {2 \left (-a e -\frac {\left (-e b +c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{3}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{3 c}\) | \(98\) |
risch | \(\frac {e \,x^{3}}{3 c}-\frac {2 \ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) a b e}{3 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) a d}{3 \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) b^{3} e}{6 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) b^{2} d}{6 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d +\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}}{6 c^{2} \left (4 a c -b^{2}\right )}-\frac {2 \ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) a b e}{3 c \left (4 a c -b^{2}\right )}+\frac {2 \ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) a d}{3 \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) b^{3} e}{6 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) b^{2} d}{6 c \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2} e +6 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d -\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, b \right ) x^{3}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c e -b^{2} e +b c d \right )^{2}}}{6 c^{2} \left (4 a c -b^{2}\right )}\) | \(1400\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 311, normalized size = 3.21 \begin {gather*} \left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} e + {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c + {\left (2 \, c x^{3} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} e + 2 \, {\left (b c d - {\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{3} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs.
\(2 (94) = 188\).
time = 96.43, size = 434, normalized size = 4.47 \begin {gather*} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) \log {\left (x^{3} + \frac {- a b e - 12 a c^{2} \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (- \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) \log {\left (x^{3} + \frac {- a b e - 12 a c^{2} \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (\frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \cdot \left (4 a c - b^{2}\right )} - \frac {b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \frac {e x^{3}}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.24, size = 95, normalized size = 0.98 \begin {gather*} \frac {x^{3} e}{3 \, c} + \frac {{\left (c d - b e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{2}} - \frac {{\left (b c d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac {2 \, c x^{3} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.95, size = 2624, normalized size = 27.05 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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