Optimal. Leaf size=398 \[ \frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt {e^2 f^2-d^2 g^2}} \]
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Rubi [A]
time = 2.84, antiderivative size = 398, 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 = {1661, 1665,
821, 739, 210} \begin {gather*} \frac {e^2 g^3 \left (13 d^2 g^2-30 d e f g+20 e^2 f^2\right ) \text {ArcTan}\left (\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2} \sqrt {e^2 f^2-d^2 g^2}}\right )}{2 (e f-d g)^2 (d g+e f)^5 \sqrt {e^2 f^2-d^2 g^2}}+\frac {3 e g^4 \sqrt {d^2-e^2 x^2} (3 e f-2 d g)}{2 (f+g x) (e f-d g)^2 (d g+e f)^5}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (f+g x)^2 (e f-d g) (d g+e f)^4}-\frac {e^2 (5 d (e f-5 d g)-e x (31 d g+e f))}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^4}+\frac {4 d e^2 (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)^3}+\frac {e^2 \left (90 d^3 g^2+e x \left (107 d^2 g^2+19 d e f g+2 e^2 f^2\right )\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 739
Rule 821
Rule 1661
Rule 1665
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {\frac {d^3 e^2 \left (e^3 f^3+15 d e^2 f^2 g+15 d^2 e f g^2+5 d^3 g^3\right )}{(e f+d g)^3}-\frac {d^2 e^3 \left (5 e^3 f^3-33 d e^2 f^2 g-45 d^2 e f g^2-15 d^3 g^3\right ) x}{(e f+d g)^3}+\frac {4 d^3 e^4 g^2 (12 e f+5 d g) x^2}{(e f+d g)^3}+\frac {16 d^3 e^5 g^3 x^3}{(e f+d g)^3}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {\frac {d^3 e^4 \left (2 e^4 f^4+17 d e^3 f^3 g+90 d^2 e^2 f^2 g^2+60 d^3 e f g^3+15 d^4 g^4\right )}{(e f+d g)^4}+\frac {3 d^3 e^5 g \left (2 e^2 f^2+45 d e f g+15 d^2 g^2\right ) x}{(e f+d g)^3}+\frac {3 d^3 e^6 g^2 \left (2 e^2 f^2+57 d e f g+25 d^2 g^2\right ) x^2}{(e f+d g)^4}+\frac {2 d^3 e^7 g^3 (e f+31 d g) x^3}{(e f+d g)^4}}{(f+g x)^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {\frac {15 d^6 e^6 g^3 \left (10 e^2 f^2+5 d e f g+d^2 g^2\right )}{(e f+d g)^5}+\frac {45 d^6 e^7 g^4 (5 e f+d g) x}{(e f+d g)^5}+\frac {90 d^6 e^8 g^5 x^2}{(e f+d g)^5}}{(f+g x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {\int \frac {\frac {30 d^6 e^7 g^3 \left (10 e^2 f^2-5 d e f g-3 d^2 g^2\right )}{(e f+d g)^4}+\frac {15 d^6 e^8 g^4 (11 e f-13 d g) x}{(e f+d g)^4}}{(f+g x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {\left (e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}-\frac {\left (e^2 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}{-e^2 f^2+d^2 g^2-x^2} \, dx,x,\frac {d^2 g+e^2 f x}{\sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5}\\ &=\frac {4 d e^2 (d+e x)}{5 (e f+d g)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {e^2 (5 d (e f-5 d g)-e (e f+31 d g) x)}{15 d (e f+d g)^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 \left (90 d^3 g^2+e \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right ) x\right )}{15 d^3 (e f+d g)^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{2 (e f-d g) (e f+d g)^4 (f+g x)^2}+\frac {3 e g^4 (3 e f-2 d g) \sqrt {d^2-e^2 x^2}}{2 (e f-d g)^2 (e f+d g)^5 (f+g x)}+\frac {e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \tan ^{-1}\left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{2 (e f-d g)^2 (e f+d g)^5 \sqrt {e^2 f^2-d^2 g^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.96, size = 387, normalized size = 0.97 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (\frac {6 e^2 (e f+d g)^2}{d (d-e x)^3}+\frac {2 e^2 (e f+d g) (2 e f+17 d g)}{d^2 (d-e x)^2}+\frac {2 e^2 \left (2 e^2 f^2+19 d e f g+107 d^2 g^2\right )}{d^3 (d-e x)}+\frac {15 g^4 (e f+d g)}{(e f-d g) (f+g x)^2}+\frac {45 e g^4 (3 e f-2 d g)}{(e f-d g)^2 (f+g x)}\right )-\frac {15 i e^2 g^3 \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) \log \left (\frac {4 (e f-d g)^2 (e f+d g)^5 \left (i d^2 g+i e^2 f x+\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}\right )}{e^2 g^2 \sqrt {e^2 f^2-d^2 g^2} \left (20 e^2 f^2-30 d e f g+13 d^2 g^2\right ) (f+g x)}\right )}{(e f-d g)^2 \sqrt {e^2 f^2-d^2 g^2}}}{30 (e f+d g)^5} \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. \(6395\) vs.
\(2(370)=740\).
time = 0.09, size = 6396, normalized size = 16.07
method | result | size |
default | \(\text {Expression too large to display}\) | \(6396\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 2615 vs.
\(2 (373) = 746\).
time = 27.97, size = 5268, normalized size = 13.24 \begin {gather*} \text {Too large to display} \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 (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (f + g x\right )^{3}}\, 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. 1314 vs.
\(2 (373) = 746\).
time = 1.41, size = 1314, normalized size = 3.30 \begin {gather*} -\frac {{\left (13 \, d^{2} g^{5} e^{2} - 30 \, d f g^{4} e^{3} + 20 \, f^{2} g^{3} e^{4}\right )} \arctan \left (\frac {d g + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f e^{\left (-1\right )}}{x}}{\sqrt {-d^{2} g^{2} + f^{2} e^{2}}}\right )}{{\left (d^{7} g^{7} + 3 \, d^{6} f g^{6} e + d^{5} f^{2} g^{5} e^{2} - 5 \, d^{4} f^{3} g^{4} e^{3} - 5 \, d^{3} f^{4} g^{3} e^{4} + d^{2} f^{5} g^{2} e^{5} + 3 \, d f^{6} g e^{6} + f^{7} e^{7}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}}} + \frac {\frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{5} g^{8} e^{\left (-4\right )}}{x^{2}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{4} f g^{7} e^{\left (-1\right )}}{x} + \frac {12 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{4} f g^{7} e^{\left (-3\right )}}{x^{2}} + \frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{4} f g^{7} e^{\left (-5\right )}}{x^{3}} + d^{3} f^{2} g^{6} e^{2} - \frac {19 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{3} f^{2} g^{6} e^{\left (-2\right )}}{x^{2}} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{3} f^{2} g^{6} e^{\left (-4\right )}}{x^{3}} + \frac {18 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{3} f^{2} g^{6}}{x} + 6 \, d^{2} f^{3} g^{5} e^{3} - \frac {29 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} f^{3} g^{5} e}{x} + \frac {6 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} f^{3} g^{5} e^{\left (-1\right )}}{x^{2}} - \frac {11 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} f^{3} g^{5} e^{\left (-3\right )}}{x^{3}} - 10 \, d f^{4} g^{4} e^{4} - \frac {10 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f^{4} g^{4}}{x^{2}}}{{\left (d^{7} f^{2} g^{7} + 3 \, d^{6} f^{3} g^{6} e + d^{5} f^{4} g^{5} e^{2} - 5 \, d^{4} f^{5} g^{4} e^{3} - 5 \, d^{3} f^{6} g^{3} e^{4} + d^{2} f^{7} g^{2} e^{5} + 3 \, d f^{8} g e^{6} + f^{9} e^{7}\right )} {\left (\frac {2 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d g e^{\left (-2\right )}}{x} + f e + \frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f e^{\left (-3\right )}}{x^{2}}\right )}^{2}} + \frac {2 \, {\left (127 \, d^{2} g^{2} e^{2} + \frac {745 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-2\right )}}{x^{2}} - \frac {525 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} g^{2} e^{\left (-4\right )}}{x^{3}} + \frac {150 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} g^{2} e^{\left (-6\right )}}{x^{4}} - \frac {485 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2}}{x} + 44 \, d f g e^{3} - \frac {145 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e}{x} + \frac {245 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-1\right )}}{x^{2}} - \frac {195 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-3\right )}}{x^{3}} + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d f g e^{\left (-5\right )}}{x^{4}} + 7 \, f^{2} e^{4} - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f^{2} e^{2}}{x} - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f^{2} e^{\left (-2\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f^{2} e^{\left (-4\right )}}{x^{4}} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f^{2}}{x^{2}}\right )}}{15 \, {\left (d^{8} g^{5} + 5 \, d^{7} f g^{4} e + 10 \, d^{6} f^{2} g^{3} e^{2} + 10 \, d^{5} f^{3} g^{2} e^{3} + 5 \, d^{4} f^{4} g e^{4} + d^{3} f^{5} e^{5}\right )} {\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 (d+e\,x\right )}^3}{{\left (f+g\,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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