Optimal. Leaf size=269 \[ \frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (5 e f+3 d g) \sqrt {d^2-e^2 x^2}}{e^6}+\frac {g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]
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Rubi [A]
time = 0.60, 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 = {1649, 1829,
655, 223, 209} \begin {gather*} -\frac {g^3 \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (13 d^2 g^2+30 d e f g+20 e^2 f^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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 655
Rule 1649
Rule 1829
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 ) \text {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 = 1.46, size = 311, normalized size = 1.16 \begin {gather*} \frac {\frac {e^3 \sqrt {d^2-e^2 x^2} \left (304 d^7 g^5+4 e^7 f^5 x^2+3 d^6 e g^4 (240 f-239 g x)-6 d e^6 f^4 x (2 f+5 g x)+2 d^2 e^5 f^3 \left (7 f^2+45 f g x+70 g^2 x^2\right )+d^5 e^2 g^3 \left (440 f^2-1710 f g x+479 g^2 x^2\right )+5 d^4 e^3 g^2 \left (8 f^3-204 f^2 g x+234 f g^2 x^2-9 g^3 x^3\right )-5 d^3 e^4 g \left (6 f^4+24 f^3 g x-128 f^2 g^2 x^2+30 f g^3 x^3+3 g^4 x^4\right )\right )}{d^3 (d-e x)^3}+15 \left (-e^2\right )^{3/2} g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{30 e^9} \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. \(1028\) vs.
\(2(247)=494\).
time = 0.14, size = 1029, normalized size = 3.83 Too large to display
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. 1489 vs.
\(2 (244) = 488\).
time = 0.52, size = 1489, normalized size = 5.54 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 781 vs.
\(2 (244) = 488\).
time = 3.08, size = 781, normalized size = 2.90 \begin {gather*} -\frac {304 \, d^{8} g^{5} - 14 \, f^{5} x^{3} e^{8} + 30 \, {\left (13 \, d^{8} g^{5} - 20 \, d^{3} f^{2} g^{3} x^{3} e^{5} - 30 \, {\left (d^{4} f g^{4} x^{3} - 2 \, d^{4} f^{2} g^{3} x^{2}\right )} e^{4} - {\left (13 \, d^{5} g^{5} x^{3} - 90 \, d^{5} f g^{4} x^{2} + 60 \, d^{5} f^{2} g^{3} x\right )} e^{3} + {\left (39 \, d^{6} g^{5} x^{2} - 90 \, d^{6} f g^{4} x + 20 \, d^{6} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (13 \, d^{7} g^{5} x - 10 \, d^{7} f g^{4}\right )} e\right )} \arctan \left (-\frac {{\left (d - \sqrt {-x^{2} e^{2} + d^{2}}\right )} e^{\left (-1\right )}}{x}\right ) + 6 \, {\left (5 \, d f^{4} g x^{3} + 7 \, d f^{5} x^{2}\right )} e^{7} - 2 \, {\left (20 \, d^{2} f^{3} g^{2} x^{3} + 45 \, d^{2} f^{4} g x^{2} + 21 \, d^{2} f^{5} x\right )} e^{6} - 2 \, {\left (220 \, d^{3} f^{2} g^{3} x^{3} - 60 \, d^{3} f^{3} g^{2} x^{2} - 45 \, d^{3} f^{4} g x - 7 \, d^{3} f^{5}\right )} e^{5} - 30 \, {\left (24 \, d^{4} f g^{4} x^{3} - 44 \, d^{4} f^{2} g^{3} x^{2} + 4 \, d^{4} f^{3} g^{2} x + d^{4} f^{4} g\right )} e^{4} - 8 \, {\left (38 \, d^{5} g^{5} x^{3} - 270 \, d^{5} f g^{4} x^{2} + 165 \, d^{5} f^{2} g^{3} x - 5 \, d^{5} f^{3} g^{2}\right )} e^{3} + 8 \, {\left (114 \, d^{6} g^{5} x^{2} - 270 \, d^{6} f g^{4} x + 55 \, d^{6} f^{2} g^{3}\right )} e^{2} - 48 \, {\left (19 \, d^{7} g^{5} x - 15 \, d^{7} f g^{4}\right )} e + {\left (304 \, d^{7} g^{5} + 4 \, f^{5} x^{2} e^{7} - 6 \, {\left (5 \, d f^{4} g x^{2} + 2 \, d f^{5} x\right )} e^{6} + 2 \, {\left (70 \, d^{2} f^{3} g^{2} x^{2} + 45 \, d^{2} f^{4} g x + 7 \, d^{2} f^{5}\right )} e^{5} - 5 \, {\left (3 \, d^{3} g^{5} x^{4} + 30 \, d^{3} f g^{4} x^{3} - 128 \, d^{3} f^{2} g^{3} x^{2} + 24 \, d^{3} f^{3} g^{2} x + 6 \, d^{3} f^{4} g\right )} e^{4} - 5 \, {\left (9 \, d^{4} g^{5} x^{3} - 234 \, d^{4} f g^{4} x^{2} + 204 \, d^{4} f^{2} g^{3} x - 8 \, d^{4} f^{3} g^{2}\right )} e^{3} + {\left (479 \, d^{5} g^{5} x^{2} - 1710 \, d^{5} f g^{4} x + 440 \, d^{5} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (239 \, d^{6} g^{5} x - 240 \, d^{6} f g^{4}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (d^{3} x^{3} e^{9} - 3 \, d^{4} x^{2} e^{8} + 3 \, d^{5} x e^{7} - d^{6} e^{6}\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 )^{5}}{\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. 934 vs.
\(2 (244) = 488\).
time = 1.33, size = 934, normalized size = 3.47 \begin {gather*} -\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}\left (d\right ) + \frac {1}{2} \, {\left (g^{5} x e^{\left (-5\right )} + 2 \, {\left (3 \, d g^{5} e^{10} + 5 \, f g^{4} e^{11}\right )} e^{\left (-16\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}} - \frac {2 \, {\left (\frac {445 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{5} g^{5} e^{\left (-2\right )}}{x} - \frac {665 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} g^{5} e^{\left (-4\right )}}{x^{2}} + \frac {405 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{5} g^{5} e^{\left (-6\right )}}{x^{3}} - \frac {90 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{5} g^{5} e^{\left (-8\right )}}{x^{4}} - 107 \, d^{5} g^{5} - 285 \, d^{4} f g^{4} e + \frac {1200 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} f g^{4} e^{\left (-1\right )}}{x} - \frac {1800 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} f g^{4} e^{\left (-3\right )}}{x^{2}} + \frac {1050 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} f g^{4} e^{\left (-5\right )}}{x^{3}} - \frac {225 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{4} f g^{4} e^{\left (-7\right )}}{x^{4}} - 220 \, d^{3} f^{2} g^{3} e^{2} - \frac {1450 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} f^{2} g^{3} e^{\left (-2\right )}}{x^{2}} + \frac {750 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} f^{2} g^{3} e^{\left (-4\right )}}{x^{3}} - \frac {150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{3} f^{2} g^{3} e^{\left (-6\right )}}{x^{4}} + \frac {950 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} f^{2} g^{3}}{x} - 20 \, d^{2} f^{3} g^{2} e^{3} + \frac {100 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f^{3} g^{2} e}{x} - \frac {200 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f^{3} g^{2} e^{\left (-1\right )}}{x^{2}} + 15 \, d f^{4} g e^{4} - \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f^{4} g e^{2}}{x} - \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f^{4} g e^{\left (-2\right )}}{x^{3}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{4} g}{x^{2}} - 7 \, f^{5} e^{5} + \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{5} e^{3}}{x} - \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{5} e}{x^{2}} + \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{5} e^{\left (-1\right )}}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{5} e^{\left (-3\right )}}{x^{4}}\right )} e^{\left (-6\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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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