Optimal. Leaf size=156 \[ \frac {d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}-\frac {4 a e \left (a e^2+5 c d^2\right )-c d x \left (15 c d^2-a e^2\right )}{48 a^3 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3} \]
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Rubi [A] time = 0.12, antiderivative size = 156, 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 = {737, 821, 778, 205} \[ -\frac {4 a e \left (a e^2+5 c d^2\right )-c d x \left (15 c d^2-a e^2\right )}{48 a^3 c^2 \left (a+c x^2\right )}+\frac {d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}}-\frac {(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac {x (d+e x)^3}{6 a \left (a+c x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 737
Rule 778
Rule 821
Rubi steps
\begin {align*} \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 {\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*}
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Mathematica [A] time = 0.15, size = 155, normalized size = 0.99 \[ \frac {\frac {\sqrt {a} \left (-4 a^4 e^3-3 a^3 c e \left (8 d^2+3 d e x+4 e^2 x^2\right )+3 a^2 c^2 d x \left (11 d^2+8 e^2 x^2\right )+a c^3 d x^3 \left (40 d^2+9 e^2 x^2\right )+15 c^4 d^3 x^5\right )}{\left (a+c x^2\right )^3}+3 \sqrt {c} d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{48 a^{7/2} c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 560, normalized size = 3.59 \[ \left [-\frac {24 \, a^{4} c e^{3} x^{2} + 48 \, a^{4} c d^{2} e + 8 \, a^{5} e^{3} - 6 \, {\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{5} - 16 \, {\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (5 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2} + {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{6} + 3 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{3} + 3 \, a^{3} c d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 6 \, {\left (11 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} x}{96 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac {12 \, a^{4} c e^{3} x^{2} + 24 \, a^{4} c d^{2} e + 4 \, a^{5} e^{3} - 3 \, {\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{5} - 8 \, {\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (5 \, a^{3} c d^{3} + 3 \, a^{4} d e^{2} + {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{6} + 3 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{3} + 3 \, a^{3} c d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 3 \, {\left (11 \, a^{3} c^{2} d^{3} - 3 \, a^{4} c d e^{2}\right )} x}{48 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 154, normalized size = 0.99 \[ \frac {{\left (5 \, c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} + \frac {15 \, c^{4} d^{3} x^{5} + 9 \, a c^{3} d x^{5} e^{2} + 40 \, a c^{3} d^{3} x^{3} + 24 \, a^{2} c^{2} d x^{3} e^{2} + 33 \, a^{2} c^{2} d^{3} x - 12 \, a^{3} c x^{2} e^{3} - 9 \, a^{3} c d x e^{2} - 24 \, a^{3} c d^{2} e - 4 \, a^{4} e^{3}}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 158, normalized size = 1.01 \[ \frac {3 d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{2} c}+\frac {5 d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}+\frac {\frac {\left (3 a \,e^{2}+5 c \,d^{2}\right ) c d \,x^{5}}{16 a^{3}}-\frac {e^{3} x^{2}}{4 c}+\frac {\left (3 a \,e^{2}+5 c \,d^{2}\right ) d \,x^{3}}{6 a^{2}}-\frac {\left (3 a \,e^{2}-11 c \,d^{2}\right ) d x}{16 a c}-\frac {\left (a \,e^{2}+6 c \,d^{2}\right ) e}{12 c^{2}}}{\left (c \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.08, size = 188, normalized size = 1.21 \[ -\frac {12 \, a^{3} c e^{3} x^{2} + 24 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 3 \, {\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{5} - 8 \, {\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (11 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{48 \, {\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )}} + \frac {{\left (5 \, c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 163, normalized size = 1.04 \[ \frac {d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,c\,d^2+3\,a\,e^2\right )}{16\,a^{7/2}\,c^{3/2}}-\frac {\frac {e^3\,x^2}{4\,c}+\frac {e\,\left (6\,c\,d^2+a\,e^2\right )}{12\,c^2}-\frac {d\,x^3\,\left (5\,c\,d^2+3\,a\,e^2\right )}{6\,a^2}+\frac {d\,x\,\left (3\,a\,e^2-11\,c\,d^2\right )}{16\,a\,c}-\frac {c\,d\,x^5\,\left (5\,c\,d^2+3\,a\,e^2\right )}{16\,a^3}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.75, size = 320, normalized size = 2.05 \[ - \frac {d \sqrt {- \frac {1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right ) \log {\left (- \frac {a^{4} c d \sqrt {- \frac {1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac {d \sqrt {- \frac {1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right ) \log {\left (\frac {a^{4} c d \sqrt {- \frac {1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac {- 4 a^{4} e^{3} - 24 a^{3} c d^{2} e - 12 a^{3} c e^{3} x^{2} + x^{5} \left (9 a c^{3} d e^{2} + 15 c^{4} d^{3}\right ) + x^{3} \left (24 a^{2} c^{2} d e^{2} + 40 a c^{3} d^{3}\right ) + x \left (- 9 a^{3} c d e^{2} + 33 a^{2} c^{2} d^{3}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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