Integrand size = 17, antiderivative size = 156 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3}-\frac {(2 a e-5 c d x) (d+e x)^2}{24 a^2 c \left (a+c x^2\right )^2}-\frac {4 a e \left (5 c d^2+a e^2\right )-c d \left (15 c d^2-a e^2\right ) x}{48 a^3 c^2 \left (a+c x^2\right )}+\frac {d \left (5 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {751, 835, 792, 211} \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\frac {d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 a e^2+5 c d^2\right )}{16 a^{7/2} c^{3/2}}-\frac {4 a e \left (a e^2+5 c d^2\right )-c d x \left (15 c d^2-a e^2\right )}{48 a^3 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3} \]
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Rule 211
Rule 751
Rule 792
Rule 835
Rubi steps \begin{align*} \text {integral}& = \frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3}-\frac {\int \frac {(-5 d-2 e x) (d+e x)^2}{\left (a+c x^2\right )^3} \, dx}{6 a} \\ & = \frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3}-\frac {(2 a e-5 c d x) (d+e x)^2}{24 a^2 c \left (a+c x^2\right )^2}-\frac {\int \frac {(d+e x) \left (-15 c d^2-4 a e^2-5 c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{24 a^2 c} \\ & = \frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3}-\frac {(2 a e-5 c d x) (d+e x)^2}{24 a^2 c \left (a+c x^2\right )^2}-\frac {4 a e \left (5 c d^2+a e^2\right )-c d \left (15 c d^2-a e^2\right ) x}{48 a^3 c^2 \left (a+c x^2\right )}+\frac {\left (d \left (5 c d^2+3 a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c} \\ & = \frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3}-\frac {(2 a e-5 c d x) (d+e x)^2}{24 a^2 c \left (a+c x^2\right )^2}-\frac {4 a e \left (5 c d^2+a e^2\right )-c d \left (15 c d^2-a e^2\right ) x}{48 a^3 c^2 \left (a+c x^2\right )}+\frac {d \left (5 c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\frac {\frac {\sqrt {a} \left (-4 a^4 e^3+15 c^4 d^3 x^5-3 a^3 c e \left (8 d^2+3 d e x+4 e^2 x^2\right )+3 a^2 c^2 d x \left (11 d^2+8 e^2 x^2\right )+a c^3 d x^3 \left (40 d^2+9 e^2 x^2\right )\right )}{\left (a+c x^2\right )^3}+3 \sqrt {c} d \left (5 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{48 a^{7/2} c^2} \]
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Time = 2.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {d \left (3 e^{2} a +5 c \,d^{2}\right ) c \,x^{5}}{16 a^{3}}+\frac {d \left (3 e^{2} a +5 c \,d^{2}\right ) x^{3}}{6 a^{2}}-\frac {e^{3} x^{2}}{4 c}-\frac {d \left (3 e^{2} a -11 c \,d^{2}\right ) x}{16 a c}-\frac {e \left (e^{2} a +6 c \,d^{2}\right )}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}+\frac {d \left (3 e^{2} a +5 c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 c \,a^{3} \sqrt {a c}}\) | \(146\) |
risch | \(\frac {\frac {d \left (3 e^{2} a +5 c \,d^{2}\right ) c \,x^{5}}{16 a^{3}}+\frac {d \left (3 e^{2} a +5 c \,d^{2}\right ) x^{3}}{6 a^{2}}-\frac {e^{3} x^{2}}{4 c}-\frac {d \left (3 e^{2} a -11 c \,d^{2}\right ) x}{16 a c}-\frac {e \left (e^{2} a +6 c \,d^{2}\right )}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}}-\frac {3 d \ln \left (c x +\sqrt {-a c}\right ) e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}-\frac {5 d^{3} \ln \left (c x +\sqrt {-a c}\right )}{32 \sqrt {-a c}\, a^{3}}+\frac {3 d \ln \left (-c x +\sqrt {-a c}\right ) e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}+\frac {5 d^{3} \ln \left (-c x +\sqrt {-a c}\right )}{32 \sqrt {-a c}\, a^{3}}\) | \(220\) |
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Time = 0.29 (sec) , antiderivative size = 560, normalized size of antiderivative = 3.59 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\left [-\frac {24 \, a^{4} c e^{3} x^{2} + 48 \, a^{4} c d^{2} e + 8 \, a^{5} e^{3} - 6 \, {\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{5} - 16 \, {\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (5 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2} + {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{6} + 3 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{3} + 3 \, a^{3} c d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 6 \, {\left (11 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} x}{96 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac {12 \, a^{4} c e^{3} x^{2} + 24 \, a^{4} c d^{2} e + 4 \, a^{5} e^{3} - 3 \, {\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{5} - 8 \, {\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (5 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2} + {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{6} + 3 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{3} + 3 \, a^{3} c d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 3 \, {\left (11 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} x}{48 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (143) = 286\).
Time = 0.94 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.05 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=- \frac {d \sqrt {- \frac {1}{a^{7} c^{3}}} \cdot \left (3 a e^{2} + 5 c d^{2}\right ) \log {\left (- \frac {a^{4} c d \sqrt {- \frac {1}{a^{7} c^{3}}} \cdot \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac {d \sqrt {- \frac {1}{a^{7} c^{3}}} \cdot \left (3 a e^{2} + 5 c d^{2}\right ) \log {\left (\frac {a^{4} c d \sqrt {- \frac {1}{a^{7} c^{3}}} \cdot \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac {- 4 a^{4} e^{3} - 24 a^{3} c d^{2} e - 12 a^{3} c e^{3} x^{2} + x^{5} \cdot \left (9 a c^{3} d e^{2} + 15 c^{4} d^{3}\right ) + x^{3} \cdot \left (24 a^{2} c^{2} d e^{2} + 40 a c^{3} d^{3}\right ) + x \left (- 9 a^{3} c d e^{2} + 33 a^{2} c^{2} d^{3}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=-\frac {12 \, a^{3} c e^{3} x^{2} + 24 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 3 \, {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{5} - 8 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (11 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (5 \, c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} \]
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Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\frac {{\left (5 \, c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} + \frac {15 \, c^{4} d^{3} x^{5} + 9 \, a c^{3} d e^{2} x^{5} + 40 \, a c^{3} d^{3} x^{3} + 24 \, a^{2} c^{2} d e^{2} x^{3} - 12 \, a^{3} c e^{3} x^{2} + 33 \, a^{2} c^{2} d^{3} x - 9 \, a^{3} c d e^{2} x - 24 \, a^{3} c d^{2} e - 4 \, a^{4} e^{3}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \]
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Time = 0.14 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^4} \, dx=\frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,c\,d^2+3\,a\,e^2\right )}{16\,a^{7/2}\,c^{3/2}}-\frac {\frac {e^3\,x^2}{4\,c}+\frac {e\,\left (6\,c\,d^2+a\,e^2\right )}{12\,c^2}-\frac {d\,x^3\,\left (5\,c\,d^2+3\,a\,e^2\right )}{6\,a^2}+\frac {d\,x\,\left (3\,a\,e^2-11\,c\,d^2\right )}{16\,a\,c}-\frac {c\,d\,x^5\,\left (5\,c\,d^2+3\,a\,e^2\right )}{16\,a^3}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \]
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