Optimal. Leaf size=242 \[ \frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}+\frac {g^3 \tan ^{-1}\left (\frac {d^2 g+e^2 f x}{\sqrt {e^2 f^2-d^2 g^2} \sqrt {d^2-e^2 x^2}}\right )}{(e f+d g)^3 \sqrt {e^2 f^2-d^2 g^2}} \]
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Rubi [A]
time = 0.40, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1661, 837, 12,
739, 210} \begin {gather*} \frac {g^3 \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 )}{(d g+e f)^3 \sqrt {e^2 f^2-d^2 g^2}}-\frac {5 d (e f-d g)-e x (11 d g+e f)}{15 d \left (d^2-e^2 x^2\right )^{3/2} (d g+e f)^2}+\frac {4 d (d+e x)}{5 \left (d^2-e^2 x^2\right )^{5/2} (d g+e f)}+\frac {15 d^3 g^2+e x \left (22 d^2 g^2+9 d e f g+2 e^2 f^2\right )}{15 d^3 \sqrt {d^2-e^2 x^2} (d g+e f)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 739
Rule 837
Rule 1661
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 {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\int \frac {\frac {d^3 e^2 (e f+5 d g)}{e f+d g}-\frac {d^2 e^3 (5 e f-11 d g) x}{e f+d g}}{(f+g x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-\frac {d^3 e^4 (e f-d g) \left (2 e^2 f^2+7 d e f g+15 d^2 g^2\right )}{e f+d g}-\frac {2 d^3 e^5 g (e f-d g) (e f+11 d g) x}{e f+d g}}{(f+g x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4 \left (e^2 f^2-d^2 g^2\right )}\\ &=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {15 d^6 e^6 g^3 (e f-d g)^2}{(e f+d g) (f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6 e^6 \left (e^2 f^2-d^2 g^2\right )^2}\\ &=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}+\frac {g^3 \int \frac {1}{(f+g x) \sqrt {d^2-e^2 x^2}} \, dx}{(e f+d g)^3}\\ &=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}-\frac {g^3 \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 )}{(e f+d g)^3}\\ &=\frac {4 d (d+e x)}{5 (e f+d g) \left (d^2-e^2 x^2\right )^{5/2}}-\frac {5 d (e f-d g)-e (e f+11 d g) x}{15 d (e f+d g)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d^3 g^2+e \left (2 e^2 f^2+9 d e f g+22 d^2 g^2\right ) x}{15 d^3 (e f+d g)^3 \sqrt {d^2-e^2 x^2}}+\frac {g^3 \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 )}{(e f+d g)^3 \sqrt {e^2 f^2-d^2 g^2}}\\ \end {align*}
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Mathematica [A]
time = 10.29, size = 225, normalized size = 0.93 \begin {gather*} \frac {\frac {\left (-e^2 f^2+d^2 g^2\right ) (d+e x) \left (32 d^4 g^2+2 e^4 f^2 x^2+3 d^3 e g (8 f-17 g x)+3 d e^3 f x (-2 f+3 g x)+d^2 e^2 \left (7 f^2-27 f g x+22 g^2 x^2\right )\right )}{d^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}}-15 g^3 \sqrt {e^2 f^2-d^2 g^2} \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 )}{15 (-e f+d g) (e f+d g)^4} \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. \(1666\) vs.
\(2(223)=446\).
time = 0.10, size = 1667, normalized size = 6.89
method | result | size |
default | \(\text {Expression too large to display}\) | \(1667\) |
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. 852 vs.
\(2 (225) = 450\).
time = 3.75, size = 1742, normalized size = 7.20 \begin {gather*} \left [\frac {32 \, d^{7} g^{4} + 7 \, f^{4} x^{3} e^{7} - 15 \, {\left (d^{3} g^{3} x^{3} e^{3} - 3 \, d^{4} g^{3} x^{2} e^{2} + 3 \, d^{5} g^{3} x e - d^{6} g^{3}\right )} \sqrt {d^{2} g^{2} - f^{2} e^{2}} \log \left (\frac {d^{3} g^{2} + d f g x e^{2} - \sqrt {d^{2} g^{2} - f^{2} e^{2}} {\left (d^{2} g + f x e^{2} + \sqrt {-x^{2} e^{2} + d^{2}} d g\right )} + {\left (d^{2} g^{2} - f^{2} e^{2}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{g x + f}\right ) + 3 \, {\left (8 \, d f^{3} g x^{3} - 7 \, d f^{4} x^{2}\right )} e^{6} + {\left (25 \, d^{2} f^{2} g^{2} x^{3} - 72 \, d^{2} f^{3} g x^{2} + 21 \, d^{2} f^{4} x\right )} e^{5} - {\left (24 \, d^{3} f g^{3} x^{3} + 75 \, d^{3} f^{2} g^{2} x^{2} - 72 \, d^{3} f^{3} g x + 7 \, d^{3} f^{4}\right )} e^{4} - {\left (32 \, d^{4} g^{4} x^{3} - 72 \, d^{4} f g^{3} x^{2} - 75 \, d^{4} f^{2} g^{2} x + 24 \, d^{4} f^{3} g\right )} e^{3} + {\left (96 \, d^{5} g^{4} x^{2} - 72 \, d^{5} f g^{3} x - 25 \, d^{5} f^{2} g^{2}\right )} e^{2} - 24 \, {\left (4 \, d^{6} g^{4} x - d^{6} f g^{3}\right )} e + {\left (32 \, d^{6} g^{4} - 2 \, f^{4} x^{2} e^{6} - 3 \, {\left (3 \, d f^{3} g x^{2} - 2 \, d f^{4} x\right )} e^{5} - {\left (20 \, d^{2} f^{2} g^{2} x^{2} - 27 \, d^{2} f^{3} g x + 7 \, d^{2} f^{4}\right )} e^{4} + 3 \, {\left (3 \, d^{3} f g^{3} x^{2} + 15 \, d^{3} f^{2} g^{2} x - 8 \, d^{3} f^{3} g\right )} e^{3} + {\left (22 \, d^{4} g^{4} x^{2} - 27 \, d^{4} f g^{3} x - 25 \, d^{4} f^{2} g^{2}\right )} e^{2} - 3 \, {\left (17 \, d^{5} g^{4} x - 8 \, d^{5} f g^{3}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{11} g^{5} + d^{3} f^{5} x^{3} e^{8} + 3 \, {\left (d^{4} f^{4} g x^{3} - d^{4} f^{5} x^{2}\right )} e^{7} + {\left (2 \, d^{5} f^{3} g^{2} x^{3} - 9 \, d^{5} f^{4} g x^{2} + 3 \, d^{5} f^{5} x\right )} e^{6} - {\left (2 \, d^{6} f^{2} g^{3} x^{3} + 6 \, d^{6} f^{3} g^{2} x^{2} - 9 \, d^{6} f^{4} g x + d^{6} f^{5}\right )} e^{5} - 3 \, {\left (d^{7} f g^{4} x^{3} - 2 \, d^{7} f^{2} g^{3} x^{2} - 2 \, d^{7} f^{3} g^{2} x + d^{7} f^{4} g\right )} e^{4} - {\left (d^{8} g^{5} x^{3} - 9 \, d^{8} f g^{4} x^{2} + 6 \, d^{8} f^{2} g^{3} x + 2 \, d^{8} f^{3} g^{2}\right )} e^{3} + {\left (3 \, d^{9} g^{5} x^{2} - 9 \, d^{9} f g^{4} x + 2 \, d^{9} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (d^{10} g^{5} x - d^{10} f g^{4}\right )} e\right )}}, \frac {32 \, d^{7} g^{4} + 7 \, f^{4} x^{3} e^{7} - 30 \, {\left (d^{3} g^{3} x^{3} e^{3} - 3 \, d^{4} g^{3} x^{2} e^{2} + 3 \, d^{5} g^{3} x e - d^{6} g^{3}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}} \arctan \left (\frac {\sqrt {-d^{2} g^{2} + f^{2} e^{2}} {\left (d g x + d f - \sqrt {-x^{2} e^{2} + d^{2}} f\right )}}{d^{2} g^{2} x - f^{2} x e^{2}}\right ) + 3 \, {\left (8 \, d f^{3} g x^{3} - 7 \, d f^{4} x^{2}\right )} e^{6} + {\left (25 \, d^{2} f^{2} g^{2} x^{3} - 72 \, d^{2} f^{3} g x^{2} + 21 \, d^{2} f^{4} x\right )} e^{5} - {\left (24 \, d^{3} f g^{3} x^{3} + 75 \, d^{3} f^{2} g^{2} x^{2} - 72 \, d^{3} f^{3} g x + 7 \, d^{3} f^{4}\right )} e^{4} - {\left (32 \, d^{4} g^{4} x^{3} - 72 \, d^{4} f g^{3} x^{2} - 75 \, d^{4} f^{2} g^{2} x + 24 \, d^{4} f^{3} g\right )} e^{3} + {\left (96 \, d^{5} g^{4} x^{2} - 72 \, d^{5} f g^{3} x - 25 \, d^{5} f^{2} g^{2}\right )} e^{2} - 24 \, {\left (4 \, d^{6} g^{4} x - d^{6} f g^{3}\right )} e + {\left (32 \, d^{6} g^{4} - 2 \, f^{4} x^{2} e^{6} - 3 \, {\left (3 \, d f^{3} g x^{2} - 2 \, d f^{4} x\right )} e^{5} - {\left (20 \, d^{2} f^{2} g^{2} x^{2} - 27 \, d^{2} f^{3} g x + 7 \, d^{2} f^{4}\right )} e^{4} + 3 \, {\left (3 \, d^{3} f g^{3} x^{2} + 15 \, d^{3} f^{2} g^{2} x - 8 \, d^{3} f^{3} g\right )} e^{3} + {\left (22 \, d^{4} g^{4} x^{2} - 27 \, d^{4} f g^{3} x - 25 \, d^{4} f^{2} g^{2}\right )} e^{2} - 3 \, {\left (17 \, d^{5} g^{4} x - 8 \, d^{5} f g^{3}\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{11} g^{5} + d^{3} f^{5} x^{3} e^{8} + 3 \, {\left (d^{4} f^{4} g x^{3} - d^{4} f^{5} x^{2}\right )} e^{7} + {\left (2 \, d^{5} f^{3} g^{2} x^{3} - 9 \, d^{5} f^{4} g x^{2} + 3 \, d^{5} f^{5} x\right )} e^{6} - {\left (2 \, d^{6} f^{2} g^{3} x^{3} + 6 \, d^{6} f^{3} g^{2} x^{2} - 9 \, d^{6} f^{4} g x + d^{6} f^{5}\right )} e^{5} - 3 \, {\left (d^{7} f g^{4} x^{3} - 2 \, d^{7} f^{2} g^{3} x^{2} - 2 \, d^{7} f^{3} g^{2} x + d^{7} f^{4} g\right )} e^{4} - {\left (d^{8} g^{5} x^{3} - 9 \, d^{8} f g^{4} x^{2} + 6 \, d^{8} f^{2} g^{3} x + 2 \, d^{8} f^{3} g^{2}\right )} e^{3} + {\left (3 \, d^{9} g^{5} x^{2} - 9 \, d^{9} f g^{4} x + 2 \, d^{9} f^{2} g^{3}\right )} e^{2} - 3 \, {\left (d^{10} g^{5} x - d^{10} f g^{4}\right )} e\right )}}\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 (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (f + g x\right )}\, 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. 607 vs.
\(2 (225) = 450\).
time = 2.00, size = 607, normalized size = 2.51 \begin {gather*} -\frac {2 \, g^{3} \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^{3} g^{3} + 3 \, d^{2} f g^{2} e + 3 \, d f^{2} g e^{2} + f^{3} e^{3}\right )} \sqrt {-d^{2} g^{2} + f^{2} e^{2}}} - \frac {2 \, {\left (\frac {115 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d^{2} g^{2} e^{\left (-2\right )}}{x} - \frac {185 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d^{2} g^{2} e^{\left (-4\right )}}{x^{2}} + \frac {135 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d^{2} g^{2} e^{\left (-6\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d^{2} g^{2} e^{\left (-8\right )}}{x^{4}} - 32 \, d^{2} g^{2} - 24 \, d f g e + \frac {75 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d f g e^{\left (-1\right )}}{x} - \frac {135 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d f g e^{\left (-3\right )}}{x^{2}} + \frac {105 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d f g e^{\left (-5\right )}}{x^{3}} - \frac {45 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} d f g e^{\left (-7\right )}}{x^{4}} - 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 )}}{15 \, {\left (d^{6} g^{3} + 3 \, d^{5} f g^{2} e + 3 \, d^{4} f^{2} g e^{2} + d^{3} f^{3} e^{3}\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 )\,{\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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