Optimal. Leaf size=117 \[ \frac {(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (2 e f-3 d g)}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {789, 653, 191} \[ \frac {(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (2 e f-3 d g)}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rule 789
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(-5 e f+3 (e f+d g)) \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {(-5 e f+3 (e f+d g)) \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(2 e f-3 d g) x}{15 d^3 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 83, normalized size = 0.71 \[ -\frac {(d+e x) \left (3 d^3 g-d^2 e (7 f+9 g x)+3 d e^2 x (2 f+g x)-2 e^3 f x^2\right )}{15 d^3 e^2 (d-e x)^2 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 183, normalized size = 1.56 \[ -\frac {7 \, d^{3} e f - 3 \, d^{4} g - {\left (7 \, e^{4} f - 3 \, d e^{3} g\right )} x^{3} + 3 \, {\left (7 \, d e^{3} f - 3 \, d^{2} e^{2} g\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{2} f - 3 \, d^{3} e g\right )} x + {\left (7 \, d^{2} e f - 3 \, d^{3} g + {\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} x^{2} - 3 \, {\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{5} x^{3} - 3 \, d^{4} e^{4} x^{2} + 3 \, d^{5} e^{3} x - d^{6} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 139, normalized size = 1.19 \[ -\frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left (15 \, d f - {\left (x {\left (\frac {{\left (3 \, d^{2} g e^{7} - 2 \, d f e^{8}\right )} x^{2} e^{\left (-4\right )}}{d^{4}} - \frac {5 \, {\left (3 \, d^{4} g e^{5} - d^{3} f e^{6}\right )} e^{\left (-4\right )}}{d^{4}}\right )} - \frac {5 \, {\left (3 \, d^{5} g e^{4} + d^{4} f e^{5}\right )} e^{\left (-4\right )}}{d^{4}}\right )} x\right )} x - \frac {{\left (3 \, d^{7} g e^{2} - 7 \, d^{6} f e^{3}\right )} e^{\left (-4\right )}}{d^{4}}\right )}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 0.73 \[ -\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (3 d \,e^{2} g \,x^{2}-2 e^{3} f \,x^{2}-9 d^{2} e g x +6 d \,e^{2} f x +3 d^{3} g -7 d^{2} e f \right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 373, normalized size = 3.19 \[ \frac {e g x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d f x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {3 \, d^{2} g x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {3 \, d^{2} f}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d^{3} g}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {4 \, f x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {g x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} + \frac {8 \, f x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}} + \frac {g x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} + \frac {{\left (e^{3} f + 3 \, d e^{2} g\right )} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} + \frac {3 \, {\left (d e^{2} f + d^{2} e g\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {2 \, {\left (e^{3} f + 3 \, d e^{2} g\right )} d^{2}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {{\left (d e^{2} f + d^{2} e g\right )} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2}} - \frac {2 \, {\left (d e^{2} f + d^{2} e g\right )} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.79, size = 79, normalized size = 0.68 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,g\,d^3-9\,g\,d^2\,e\,x-7\,f\,d^2\,e+3\,g\,d\,e^2\,x^2+6\,f\,d\,e^2\,x-2\,f\,e^3\,x^2\right )}{15\,d^3\,e^2\,{\left (d-e\,x\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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