Optimal. Leaf size=340 \[ \frac {5 (2 c d-b e)^3 (-7 b e g+6 c d g+8 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2}-\frac {5 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{64 c^4 e^2}-\frac {5 (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{96 c^3 e^2}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2} \]
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Rubi [A] time = 0.61, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {794, 670, 640, 621, 204} \[ -\frac {5 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{64 c^4 e^2}-\frac {5 (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{96 c^3 e^2}-\frac {(d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+6 c d g+8 c e f)}{24 c^2 e^2}+\frac {5 (2 c d-b e)^3 (-7 b e g+6 c d g+8 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}-\frac {\left (\frac {1}{2} e \left (-2 c e^2 f+b e^2 g\right )+3 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right ) \int \frac {(d+e x)^3}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 c e^3}\\ &=-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac {(5 (2 c d-b e) (8 c e f+6 c d g-7 b e g)) \int \frac {(d+e x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{48 c^2 e}\\ &=-\frac {5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac {\left (5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g)\right ) \int \frac {d+e x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{64 c^3 e}\\ &=-\frac {5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac {5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac {\left (5 (2 c d-b e)^3 (8 c e f+6 c d g-7 b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{128 c^4 e}\\ &=-\frac {5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac {5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac {\left (5 (2 c d-b e)^3 (8 c e f+6 c d g-7 b e g)\right ) \operatorname {Subst}\left (\int \frac {1}{-4 c e^2-x^2} \, dx,x,\frac {-b e^2-2 c e^2 x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{64 c^4 e}\\ &=-\frac {5 (2 c d-b e)^2 (8 c e f+6 c d g-7 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{64 c^4 e^2}-\frac {5 (2 c d-b e) (8 c e f+6 c d g-7 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{96 c^3 e^2}-\frac {(8 c e f+6 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^2}-\frac {g (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^2}+\frac {5 (2 c d-b e)^3 (8 c e f+6 c d g-7 b e g) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{9/2} e^2}\\ \end {align*}
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Mathematica [A] time = 1.74, size = 375, normalized size = 1.10 \[ \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (-\frac {e^3 (-7 b e g+6 c d g+8 c e f) \left (8 c^3 e^6 (d+e x)^3 \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}-10 c^2 e^6 (d+e x)^2 \sqrt {e (2 c d-b e)} (b e-2 c d) \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}+15 \sqrt {c} e^{13/2} \sqrt {d+e x} (b e-2 c d)^3 \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )+15 c e^6 (d+e x) \sqrt {e (2 c d-b e)} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}\right )}{3 \sqrt {e (2 c d-b e)} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}-16 c^4 e^9 g (d+e x)^4\right )}{64 c^5 e^{11} (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.24, size = 825, normalized size = 2.43 \[ \left [-\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f + {\left (48 \, c^{4} d^{4} - 128 \, b c^{3} d^{3} e + 120 \, b^{2} c^{2} d^{2} e^{2} - 48 \, b^{3} c d e^{3} + 7 \, b^{4} e^{4}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f + {\left (24 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 70 \, b c^{3} d e^{2} + 15 \, b^{2} c^{2} e^{3}\right )} f + {\left (576 \, c^{4} d^{3} - 1036 \, b c^{3} d^{2} e + 580 \, b^{2} c^{2} d e^{2} - 105 \, b^{3} c e^{3}\right )} g + 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 5 \, b c^{3} e^{3}\right )} f + {\left (180 \, c^{4} d^{2} e - 156 \, b c^{3} d e^{2} + 35 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{768 \, c^{5} e^{2}}, -\frac {15 \, {\left (8 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} f + {\left (48 \, c^{4} d^{4} - 128 \, b c^{3} d^{3} e + 120 \, b^{2} c^{2} d^{2} e^{2} - 48 \, b^{3} c d e^{3} + 7 \, b^{4} e^{4}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (48 \, c^{4} e^{3} g x^{3} + 8 \, {\left (8 \, c^{4} e^{3} f + {\left (24 \, c^{4} d e^{2} - 7 \, b c^{3} e^{3}\right )} g\right )} x^{2} + 8 \, {\left (88 \, c^{4} d^{2} e - 70 \, b c^{3} d e^{2} + 15 \, b^{2} c^{2} e^{3}\right )} f + {\left (576 \, c^{4} d^{3} - 1036 \, b c^{3} d^{2} e + 580 \, b^{2} c^{2} d e^{2} - 105 \, b^{3} c e^{3}\right )} g + 2 \, {\left (8 \, {\left (18 \, c^{4} d e^{2} - 5 \, b c^{3} e^{3}\right )} f + {\left (180 \, c^{4} d^{2} e - 156 \, b c^{3} d e^{2} + 35 \, b^{2} c^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{384 \, c^{5} e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 383, normalized size = 1.13 \[ -\frac {1}{192} \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (2 \, {\left (4 \, {\left (\frac {6 \, g x e}{c} + \frac {{\left (24 \, c^{3} d g e^{4} + 8 \, c^{3} f e^{5} - 7 \, b c^{2} g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} x + \frac {{\left (180 \, c^{3} d^{2} g e^{3} + 144 \, c^{3} d f e^{4} - 156 \, b c^{2} d g e^{4} - 40 \, b c^{2} f e^{5} + 35 \, b^{2} c g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} x + \frac {{\left (576 \, c^{3} d^{3} g e^{2} + 704 \, c^{3} d^{2} f e^{3} - 1036 \, b c^{2} d^{2} g e^{3} - 560 \, b c^{2} d f e^{4} + 580 \, b^{2} c d g e^{4} + 120 \, b^{2} c f e^{5} - 105 \, b^{3} g e^{5}\right )} e^{\left (-4\right )}}{c^{4}}\right )} + \frac {5 \, {\left (48 \, c^{4} d^{4} g + 64 \, c^{4} d^{3} f e - 128 \, b c^{3} d^{3} g e - 96 \, b c^{3} d^{2} f e^{2} + 120 \, b^{2} c^{2} d^{2} g e^{2} + 48 \, b^{2} c^{2} d f e^{3} - 48 \, b^{3} c d g e^{3} - 8 \, b^{3} c f e^{4} + 7 \, b^{4} g e^{4}\right )} \sqrt {-c e^{2}} e^{\left (-3\right )} \log \left ({\left | -2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}\right )} c - \sqrt {-c e^{2}} b \right |}\right )}{128 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1208, normalized size = 3.55 \[ \frac {35 b^{4} e^{3} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{128 \sqrt {c \,e^{2}}\, c^{4}}-\frac {15 b^{3} d \,e^{2} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, c^{3}}-\frac {5 b^{3} e^{3} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{16 \sqrt {c \,e^{2}}\, c^{3}}+\frac {75 b^{2} d^{2} e g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{16 \sqrt {c \,e^{2}}\, c^{2}}+\frac {15 b^{2} d \,e^{2} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, c^{2}}-\frac {5 b \,d^{3} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}\, c}-\frac {15 b \,d^{2} e f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{4 \sqrt {c \,e^{2}}\, c}+\frac {15 d^{4} g \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{8 \sqrt {c \,e^{2}}\, e}+\frac {5 d^{3} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}\right )}{2 \sqrt {c \,e^{2}}}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, e g \,x^{3}}{4 c}+\frac {7 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b e g \,x^{2}}{24 c^{2}}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d g \,x^{2}}{c}-\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, e f \,x^{2}}{3 c}-\frac {35 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} e g x}{96 c^{3}}+\frac {13 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b d g x}{8 c^{2}}+\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b e f x}{12 c^{2}}-\frac {15 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d^{2} g x}{8 c e}-\frac {3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d f x}{2 c}+\frac {35 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{3} e g}{64 c^{4}}-\frac {145 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} d g}{48 c^{3}}-\frac {5 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b^{2} e f}{8 c^{3}}+\frac {259 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b \,d^{2} g}{48 c^{2} e}+\frac {35 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, b d f}{12 c^{2}}-\frac {3 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d^{3} g}{c \,e^{2}}-\frac {11 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, d^{2} f}{3 c e} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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