Optimal. Leaf size=269 \[ -\frac {g^3 \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}+\frac {g^4 \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e^6}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.97, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1635, 1815, 641, 217, 203} \[ \frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}+\frac {g^4 \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e^6}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 641
Rule 1635
Rule 1815
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (-\frac {2 e^5 f^5-15 d e^4 f^4 g-30 d^2 e^3 f^3 g^2-30 d^3 e^2 f^2 g^3-15 d^4 e f g^4-3 d^5 g^5}{e^5}+\frac {5 d g^2 \left (10 e^3 f^3+10 d e^2 f^2 g+5 d^2 e f g^2+d^3 g^3\right ) x}{e^4}+\frac {5 d g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right ) x^2}{e^3}+\frac {5 d g^4 (5 e f+d g) x^3}{e^2}+\frac {5 d g^5 x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {2 e^5 f^5-15 d e^4 f^4 g+70 d^2 e^3 f^3 g^2+170 d^3 e^2 f^2 g^3+135 d^4 e f g^4+37 d^5 g^5}{e^5}+\frac {15 d^2 g^3 \left (10 e^2 f^2+10 d e f g+3 d^2 g^2\right ) x}{e^4}+\frac {15 d^2 g^4 (5 e f+2 d g) x^2}{e^3}+\frac {15 d^2 g^5 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\frac {15 d^3 g^3 \left (10 e^2 f^2+15 d e f g+6 d^2 g^2\right )}{e^5}+\frac {15 d^3 g^4 (5 e f+3 d g) x}{e^4}+\frac {15 d^3 g^5 x^2}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {\int \frac {-\frac {15 d^3 g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )}{e^3}-\frac {30 d^3 g^4 (5 e f+3 d g) x}{e^2}}{\sqrt {d^2-e^2 x^2}} \, dx}{30 d^3 e^2}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^5}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {\left (g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5}\\ &=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}\\ \end {align*}
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Mathematica [A] time = 0.97, size = 193, normalized size = 0.72 \[ \frac {\sqrt {d^2-e^2 x^2} \left (\frac {2 (2 e f-23 d g) (d g+e f)^4}{d^2 (d-e x)^2}+\frac {2 (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d^3 (d-e x)}+30 g^4 (3 d g+5 e f)+\frac {6 (d g+e f)^5}{d (d-e x)^3}+15 e g^5 x\right )-15 g^3 \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{30 e^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 807, normalized size = 3.00 \[ -\frac {14 \, d^{3} e^{5} f^{5} - 30 \, d^{4} e^{4} f^{4} g + 40 \, d^{5} e^{3} f^{3} g^{2} + 440 \, d^{6} e^{2} f^{2} g^{3} + 720 \, d^{7} e f g^{4} + 304 \, d^{8} g^{5} - 2 \, {\left (7 \, e^{8} f^{5} - 15 \, d e^{7} f^{4} g + 20 \, d^{2} e^{6} f^{3} g^{2} + 220 \, d^{3} e^{5} f^{2} g^{3} + 360 \, d^{4} e^{4} f g^{4} + 152 \, d^{5} e^{3} g^{5}\right )} x^{3} + 6 \, {\left (7 \, d e^{7} f^{5} - 15 \, d^{2} e^{6} f^{4} g + 20 \, d^{3} e^{5} f^{3} g^{2} + 220 \, d^{4} e^{4} f^{2} g^{3} + 360 \, d^{5} e^{3} f g^{4} + 152 \, d^{6} e^{2} g^{5}\right )} x^{2} - 6 \, {\left (7 \, d^{2} e^{6} f^{5} - 15 \, d^{3} e^{5} f^{4} g + 20 \, d^{4} e^{4} f^{3} g^{2} + 220 \, d^{5} e^{3} f^{2} g^{3} + 360 \, d^{6} e^{2} f g^{4} + 152 \, d^{7} e g^{5}\right )} x + 30 \, {\left (20 \, d^{6} e^{2} f^{2} g^{3} + 30 \, d^{7} e f g^{4} + 13 \, d^{8} g^{5} - {\left (20 \, d^{3} e^{5} f^{2} g^{3} + 30 \, d^{4} e^{4} f g^{4} + 13 \, d^{5} e^{3} g^{5}\right )} x^{3} + 3 \, {\left (20 \, d^{4} e^{4} f^{2} g^{3} + 30 \, d^{5} e^{3} f g^{4} + 13 \, d^{6} e^{2} g^{5}\right )} x^{2} - 3 \, {\left (20 \, d^{5} e^{3} f^{2} g^{3} + 30 \, d^{6} e^{2} f g^{4} + 13 \, d^{7} e g^{5}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, d^{3} e^{4} g^{5} x^{4} - 14 \, d^{2} e^{5} f^{5} + 30 \, d^{3} e^{4} f^{4} g - 40 \, d^{4} e^{3} f^{3} g^{2} - 440 \, d^{5} e^{2} f^{2} g^{3} - 720 \, d^{6} e f g^{4} - 304 \, d^{7} g^{5} + 15 \, {\left (10 \, d^{3} e^{4} f g^{4} + 3 \, d^{4} e^{3} g^{5}\right )} x^{3} - {\left (4 \, e^{7} f^{5} - 30 \, d e^{6} f^{4} g + 140 \, d^{2} e^{5} f^{3} g^{2} + 640 \, d^{3} e^{4} f^{2} g^{3} + 1170 \, d^{4} e^{3} f g^{4} + 479 \, d^{5} e^{2} g^{5}\right )} x^{2} + 3 \, {\left (4 \, d e^{6} f^{5} - 30 \, d^{2} e^{5} f^{4} g + 40 \, d^{3} e^{4} f^{3} g^{2} + 340 \, d^{4} e^{3} f^{2} g^{3} + 570 \, d^{5} e^{2} f g^{4} + 239 \, d^{6} e g^{5}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{3} e^{9} x^{3} - 3 \, d^{4} e^{8} x^{2} + 3 \, d^{5} e^{7} x - d^{6} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 537, normalized size = 2.00 \[ -\frac {1}{2} \, {\left (13 \, d^{2} g^{5} + 30 \, d f g^{4} e + 20 \, f^{2} g^{3} e^{2}\right )} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\relax (d) + \frac {\sqrt {-x^{2} e^{2} + d^{2}} {\left ({\left ({\left ({\left ({\left ({\left (15 \, {\left (g^{5} x e + \frac {2 \, {\left (3 \, d^{5} g^{5} e^{12} + 5 \, d^{4} f g^{4} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {{\left (299 \, d^{6} g^{5} e^{11} + 720 \, d^{5} f g^{4} e^{12} + 640 \, d^{4} f^{2} g^{3} e^{13} + 140 \, d^{3} f^{3} g^{2} e^{14} - 30 \, d^{2} f^{4} g e^{15} + 4 \, d f^{5} e^{16}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {30 \, {\left (19 \, d^{7} g^{5} e^{10} + 45 \, d^{6} f g^{4} e^{11} + 30 \, d^{5} f^{2} g^{3} e^{12} + 10 \, d^{4} f^{3} g^{2} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x + \frac {5 \, {\left (91 \, d^{8} g^{5} e^{9} + 210 \, d^{7} f g^{4} e^{10} + 140 \, d^{6} f^{2} g^{3} e^{11} - 20 \, d^{5} f^{3} g^{2} e^{12} - 30 \, d^{4} f^{4} g e^{13} + 2 \, d^{3} f^{5} e^{14}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x + \frac {10 \, {\left (76 \, d^{9} g^{5} e^{8} + 180 \, d^{8} f g^{4} e^{9} + 110 \, d^{7} f^{2} g^{3} e^{10} + 10 \, d^{6} f^{3} g^{2} e^{11} - 15 \, d^{5} f^{4} g e^{12} - d^{4} f^{5} e^{13}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {15 \, {\left (13 \, d^{10} g^{5} e^{7} + 30 \, d^{9} f g^{4} e^{8} + 20 \, d^{8} f^{2} g^{3} e^{9} + 2 \, d^{5} f^{5} e^{12}\right )} e^{\left (-12\right )}}{d^{4}}\right )} x - \frac {2 \, {\left (152 \, d^{11} g^{5} e^{6} + 360 \, d^{10} f g^{4} e^{7} + 220 \, d^{9} f^{2} g^{3} e^{8} + 20 \, d^{8} f^{3} g^{2} e^{9} - 15 \, d^{7} f^{4} g e^{10} + 7 \, d^{6} f^{5} e^{11}\right )} e^{\left (-12\right )}}{d^{4}}\right )}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1308, normalized size = 4.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.05, size = 1579, normalized size = 5.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (f+g\,x\right )}^5\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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