Optimal. Leaf size=145 \[ \frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1649, 803, 651}
\begin {gather*} \frac {(d+e x)^3 (d g+e f)^2}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x)^2 (e f-4 d g) (d g+e f)}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x) \left (7 d^2 g^2-6 d e f g+2 e^2 f^2\right )}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 803
Rule 1649
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-2 f^2+\frac {6 d f g}{e}+\frac {3 d^2 g^2}{e^2}+\frac {5 d g^2 x}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) \int \frac {d+e x}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2 e^2}\\ &=\frac {(e f+d g)^2 (d+e x)^3}{5 d e^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-4 d g) (e f+d g) (d+e x)^2}{15 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\left (2 e^2 f^2-6 d e f g+7 d^2 g^2\right ) (d+e x)}{15 d^3 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.61, size = 105, normalized size = 0.72 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (2 d^4 g^2+2 e^4 f^2 x^2-6 d^3 e g (f+g x)-6 d e^3 f x (f+g x)+d^2 e^2 \left (7 f^2+18 f g x+7 g^2 x^2\right )\right )}{15 d^3 e^3 (d-e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(531\) vs.
\(2(133)=266\).
time = 0.09, size = 532, normalized size = 3.67
method | result | size |
trager | \(\frac {\left (7 d^{2} e^{2} g^{2} x^{2}-6 d \,e^{3} f g \,x^{2}+2 e^{4} f^{2} x^{2}-6 d^{3} e \,g^{2} x +18 d^{2} e^{2} f g x -6 d \,e^{3} f^{2} x +2 d^{4} g^{2}-6 f g \,d^{3} e +7 d^{2} e^{2} f^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} e^{3} \left (-e x +d \right )^{3}}\) | \(126\) |
gosper | \(\frac {\left (e x +d \right )^{4} \left (-e x +d \right ) \left (7 d^{2} e^{2} g^{2} x^{2}-6 d \,e^{3} f g \,x^{2}+2 e^{4} f^{2} x^{2}-6 d^{3} e \,g^{2} x +18 d^{2} e^{2} f g x -6 d \,e^{3} f^{2} x +2 d^{4} g^{2}-6 f g \,d^{3} e +7 d^{2} e^{2} f^{2}\right )}{15 d^{3} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(131\) |
default | \(g^{2} e^{3} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+\left (3 e^{2} d \,g^{2}+2 e^{3} f g \right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\left (3 g^{2} d^{2} e +6 f g d \,e^{2}+f^{2} e^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+\left (d^{3} g^{2}+6 d^{2} e f g +3 d \,e^{2} f^{2}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {2 d^{3} f g +3 e \,d^{2} f^{2}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{3} f^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\) | \(532\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 544 vs.
\(2 (134) = 268\).
time = 0.32, size = 544, normalized size = 3.75 \begin {gather*} \frac {g^{2} x^{4} e}{{\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {4 \, d^{2} g^{2} x^{2} e^{\left (-1\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {8 \, d^{4} g^{2} e^{\left (-3\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {2 \, d^{3} f g e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f^{2} e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x^{3} e^{\left (-2\right )}}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {3 \, {\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} d^{2} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d f^{2} x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d^{2} g^{2} e + 6 \, d f g e^{2} + f^{2} e^{3}\right )} x^{2} e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (3 \, d^{2} g^{2} e + 6 \, d f g e^{2} + f^{2} e^{3}\right )} d^{2} e^{\left (-4\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x e^{\left (-4\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {4 \, f^{2} x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} + \frac {{\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {{\left (3 \, d g^{2} e^{2} + 2 \, f g e^{3}\right )} x e^{\left (-4\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} + \frac {8 \, f^{2} x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} - \frac {{\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} - \frac {2 \, {\left (d^{3} g^{2} + 6 \, d^{2} f g e + 3 \, d f^{2} e^{2}\right )} x e^{\left (-2\right )}}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 266, normalized size = 1.83 \begin {gather*} -\frac {2 \, d^{5} g^{2} - 7 \, f^{2} x^{3} e^{5} + 3 \, {\left (2 \, d f g x^{3} + 7 \, d f^{2} x^{2}\right )} e^{4} - {\left (2 \, d^{2} g^{2} x^{3} + 18 \, d^{2} f g x^{2} + 21 \, d^{2} f^{2} x\right )} e^{3} + {\left (6 \, d^{3} g^{2} x^{2} + 18 \, d^{3} f g x + 7 \, d^{3} f^{2}\right )} e^{2} - 6 \, {\left (d^{4} g^{2} x + d^{4} f g\right )} e + {\left (2 \, d^{4} g^{2} + 2 \, f^{2} x^{2} e^{4} - 6 \, {\left (d f g x^{2} + d f^{2} x\right )} e^{3} + {\left (7 \, d^{2} g^{2} x^{2} + 18 \, d^{2} f g x + 7 \, d^{2} f^{2}\right )} e^{2} - 6 \, {\left (d^{3} g^{2} x + d^{3} f g\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{6} - 3 \, d^{4} x^{2} e^{5} + 3 \, d^{5} x e^{4} - d^{6} e^{3}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 356 vs.
\(2 (134) = 268\).
time = 1.99, size = 356, normalized size = 2.46 \begin {gather*} -\frac {2 \, {\left (\frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2} e^{\left (-2\right )}}{x} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-4\right )}}{x^{2}} - 2 \, d^{2} g^{2} + 6 \, d f g e - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e^{\left (-1\right )}}{x} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-3\right )}}{x^{2}} - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-5\right )}}{x^{3}} - 7 \, f^{2} e^{2} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{2} e^{\left (-2\right )}}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{2} e^{\left (-4\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{2} e^{\left (-6\right )}}{x^{4}} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{2}}{x}\right )} e^{\left (-3\right )}}{15 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.87, size = 125, normalized size = 0.86 \begin {gather*} \frac {\sqrt {d^2-e^2\,x^2}\,\left (2\,d^4\,g^2-6\,d^3\,e\,f\,g-6\,d^3\,e\,g^2\,x+7\,d^2\,e^2\,f^2+18\,d^2\,e^2\,f\,g\,x+7\,d^2\,e^2\,g^2\,x^2-6\,d\,e^3\,f^2\,x-6\,d\,e^3\,f\,g\,x^2+2\,e^4\,f^2\,x^2\right )}{15\,d^3\,e^3\,{\left (d-e\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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