Optimal. Leaf size=155 \[ \frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {751, 819, 737,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{16 a^{7/2} c^{5/2}}-\frac {(d+e x) \left (a e^2+5 c d^2\right ) (a e-c d x)}{16 a^3 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 737
Rule 751
Rule 819
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^4} \, dx &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {\int \frac {(-5 d-e x) (d+e x)^3}{\left (a+c x^2\right )^3} \, dx}{6 a}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c^2}\\ &=\frac {x (d+e x)^4}{6 a \left (a+c x^2\right )^3}-\frac {(a e-5 c d x) (d+e x)^3}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\left (5 c d^2+a e^2\right ) (a e-c d x) (d+e x)}{16 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 197, normalized size = 1.27 \begin {gather*} \frac {15 c^4 d^4 x^5-a^4 e^3 (16 d+3 e x)+2 a c^3 d^2 x^3 \left (20 d^2+9 e^2 x^2\right )-2 a^3 c e \left (16 d^3+9 d^2 e x+24 d e^2 x^2+4 e^3 x^3\right )+3 a^2 c^2 x \left (11 d^4+16 d^2 e^2 x^2+e^4 x^4\right )}{48 a^3 c^2 \left (a+c x^2\right )^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 198, normalized size = 1.28
method | result | size |
default | \(\frac {\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{5}}{16 a^{3}}-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right ) x^{3}}{6 a^{2} c}-\frac {d \,e^{3} x^{2}}{c}-\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{16 c^{2} a}-\frac {d e \left (e^{2} a +2 c \,d^{2}\right )}{3 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}+\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 a^{3} c^{2} \sqrt {a c}}\) | \(198\) |
risch | \(\frac {\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) x^{5}}{16 a^{3}}-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}-5 c^{2} d^{4}\right ) x^{3}}{6 a^{2} c}-\frac {d \,e^{3} x^{2}}{c}-\frac {\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}-11 c^{2} d^{4}\right ) x}{16 c^{2} a}-\frac {d e \left (e^{2} a +2 c \,d^{2}\right )}{3 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) e^{4}}{32 \sqrt {-a c}\, c^{2} a}-\frac {3 \ln \left (c x +\sqrt {-a c}\right ) d^{2} e^{2}}{16 \sqrt {-a c}\, c \,a^{2}}-\frac {5 \ln \left (c x +\sqrt {-a c}\right ) d^{4}}{32 \sqrt {-a c}\, a^{3}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) e^{4}}{32 \sqrt {-a c}\, c^{2} a}+\frac {3 \ln \left (-c x +\sqrt {-a c}\right ) d^{2} e^{2}}{16 \sqrt {-a c}\, c \,a^{2}}+\frac {5 \ln \left (-c x +\sqrt {-a c}\right ) d^{4}}{32 \sqrt {-a c}\, a^{3}}\) | \(321\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 226, normalized size = 1.46 \begin {gather*} -\frac {48 \, a^{3} c d x^{2} e^{3} + 32 \, a^{3} c d^{3} e - 3 \, {\left (5 \, c^{4} d^{4} + 6 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{5} + 16 \, a^{4} d e^{3} - 8 \, {\left (5 \, a c^{3} d^{4} + 6 \, a^{2} c^{2} d^{2} e^{2} - a^{3} c e^{4}\right )} x^{3} - 3 \, {\left (11 \, a^{2} c^{2} d^{4} - 6 \, a^{3} c d^{2} e^{2} - a^{4} e^{4}\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 353 vs.
\(2 (142) = 284\).
time = 1.69, size = 727, normalized size = 4.69 \begin {gather*} \left [\frac {30 \, a c^{5} d^{4} x^{5} + 80 \, a^{2} c^{4} d^{4} x^{3} + 66 \, a^{3} c^{3} d^{4} x - 64 \, a^{4} c^{2} d^{3} e - 3 \, {\left (5 \, c^{5} d^{4} x^{6} + 15 \, a c^{4} d^{4} x^{4} + 15 \, a^{2} c^{3} d^{4} x^{2} + 5 \, a^{3} c^{2} d^{4} + {\left (a^{2} c^{3} x^{6} + 3 \, a^{3} c^{2} x^{4} + 3 \, a^{4} c x^{2} + a^{5}\right )} e^{4} + 6 \, {\left (a c^{4} d^{2} x^{6} + 3 \, a^{2} c^{3} d^{2} x^{4} + 3 \, a^{3} c^{2} d^{2} x^{2} + a^{4} c d^{2}\right )} e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (3 \, a^{3} c^{3} x^{5} - 8 \, a^{4} c^{2} x^{3} - 3 \, a^{5} c x\right )} e^{4} - 32 \, {\left (3 \, a^{4} c^{2} d x^{2} + a^{5} c d\right )} e^{3} + 12 \, {\left (3 \, a^{2} c^{4} d^{2} x^{5} + 8 \, a^{3} c^{3} d^{2} x^{3} - 3 \, a^{4} c^{2} d^{2} x\right )} e^{2}}{96 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} c^{3}\right )}}, \frac {15 \, a c^{5} d^{4} x^{5} + 40 \, a^{2} c^{4} d^{4} x^{3} + 33 \, a^{3} c^{3} d^{4} x - 32 \, a^{4} c^{2} d^{3} e + 3 \, {\left (5 \, c^{5} d^{4} x^{6} + 15 \, a c^{4} d^{4} x^{4} + 15 \, a^{2} c^{3} d^{4} x^{2} + 5 \, a^{3} c^{2} d^{4} + {\left (a^{2} c^{3} x^{6} + 3 \, a^{3} c^{2} x^{4} + 3 \, a^{4} c x^{2} + a^{5}\right )} e^{4} + 6 \, {\left (a c^{4} d^{2} x^{6} + 3 \, a^{2} c^{3} d^{2} x^{4} + 3 \, a^{3} c^{2} d^{2} x^{2} + a^{4} c d^{2}\right )} e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (3 \, a^{3} c^{3} x^{5} - 8 \, a^{4} c^{2} x^{3} - 3 \, a^{5} c x\right )} e^{4} - 16 \, {\left (3 \, a^{4} c^{2} d x^{2} + a^{5} c d\right )} e^{3} + 6 \, {\left (3 \, a^{2} c^{4} d^{2} x^{5} + 8 \, a^{3} c^{3} d^{2} x^{3} - 3 \, a^{4} c^{2} d^{2} x\right )} e^{2}}{48 \, {\left (a^{4} c^{6} x^{6} + 3 \, a^{5} c^{5} x^{4} + 3 \, a^{6} c^{4} x^{2} + a^{7} 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. 413 vs.
\(2 (144) = 288\).
time = 1.93, size = 413, normalized size = 2.66 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (- \frac {a^{4} c^{2} \sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right ) \log {\left (\frac {a^{4} c^{2} \sqrt {- \frac {1}{a^{7} c^{5}}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{a^{2} e^{4} + 6 a c d^{2} e^{2} + 5 c^{2} d^{4}} + x \right )}}{32} + \frac {- 16 a^{4} d e^{3} - 32 a^{3} c d^{3} e - 48 a^{3} c d e^{3} x^{2} + x^{5} \cdot \left (3 a^{2} c^{2} e^{4} + 18 a c^{3} d^{2} e^{2} + 15 c^{4} d^{4}\right ) + x^{3} \left (- 8 a^{3} c e^{4} + 48 a^{2} c^{2} d^{2} e^{2} + 40 a c^{3} d^{4}\right ) + x \left (- 3 a^{4} e^{4} - 18 a^{3} c d^{2} e^{2} + 33 a^{2} c^{2} d^{4}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 205, normalized size = 1.32 \begin {gather*} \frac {{\left (5 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c^{2}} + \frac {15 \, c^{4} d^{4} x^{5} + 18 \, a c^{3} d^{2} x^{5} e^{2} + 40 \, a c^{3} d^{4} x^{3} + 3 \, a^{2} c^{2} x^{5} e^{4} + 48 \, a^{2} c^{2} d^{2} x^{3} e^{2} + 33 \, a^{2} c^{2} d^{4} x - 8 \, a^{3} c x^{3} e^{4} - 48 \, a^{3} c d x^{2} e^{3} - 18 \, a^{3} c d^{2} x e^{2} - 32 \, a^{3} c d^{3} e - 3 \, a^{4} x e^{4} - 16 \, a^{4} d e^{3}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.44, size = 263, normalized size = 1.70 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x\,\left (c\,d^2+a\,e^2\right )\,\left (5\,c\,d^2+a\,e^2\right )}{\sqrt {a}\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}\right )\,\left (c\,d^2+a\,e^2\right )\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^{7/2}\,c^{5/2}}-\frac {\frac {d\,e^3\,x^2}{c}-\frac {x^5\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{16\,a^3}+\frac {x\,\left (a^2\,e^4+6\,a\,c\,d^2\,e^2-11\,c^2\,d^4\right )}{16\,a\,c^2}+\frac {d\,e\,\left (2\,c\,d^2+a\,e^2\right )}{3\,c^2}-\frac {x^3\,\left (-a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{6\,a^2\,c}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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