Optimal. Leaf size=287 \[ -\frac {3 (2 c d-b e) (-5 b e g+6 c d g+4 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{7/2} e^2}+\frac {3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{4 c^3 e^2}+\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{2 c^2 e^2 (2 c d-b e)}+\frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.44, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {788, 670, 640, 621, 204} \begin {gather*} \frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{2 c^2 e^2 (2 c d-b e)}+\frac {3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+6 c d g+4 c e f)}{4 c^3 e^2}-\frac {3 (2 c d-b e) (-5 b e g+6 c d g+4 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 )}{8 c^{7/2} e^2}+\frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 640
Rule 670
Rule 788
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(4 c e f+6 c d g-5 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}{c e (2 c d-b e)}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {(4 c e f+6 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^2 e^2 (2 c d-b e)}-\frac {(3 (4 c e f+6 c d g-5 b e g)) \int \frac {d+e x}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 c^2 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^3 e^2}+\frac {(4 c e f+6 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^2 e^2 (2 c d-b e)}-\frac {(3 (2 c d-b e) (4 c e f+6 c d g-5 b e g)) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{8 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^3 e^2}+\frac {(4 c e f+6 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^2 e^2 (2 c d-b e)}-\frac {(3 (2 c d-b e) (4 c e f+6 c d g-5 b e g)) \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 )}{4 c^3 e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^3 e^2}+\frac {(4 c e f+6 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^2 e^2 (2 c d-b e)}-\frac {3 (2 c d-b e) (4 c e f+6 c d g-5 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 )}{8 c^{7/2} e^2}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 250, normalized size = 0.87 \begin {gather*} \frac {\sqrt {c} \sqrt {e} (d+e x) \sqrt {e (2 c d-b e)} \left (15 b^2 e^2 g+b c e (-43 d g-12 e f+5 e g x)+2 c^2 \left (14 d^2 g+5 d e (2 f-g x)-e^2 x (2 f+g x)\right )\right )-3 e \sqrt {d+e x} (b e-2 c d)^2 \sqrt {\frac {b e-c d+c e x}{b e-2 c d}} (-5 b e g+6 c d g+4 c e f) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{4 c^{7/2} e^{5/2} \sqrt {e (2 c d-b e)} \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.19, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.33, size = 745, normalized size = 2.60 \begin {gather*} \left [\frac {3 \, {\left (4 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - {\left (4 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\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 (2 \, c^{3} e^{2} g x^{2} - 4 \, {\left (5 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f - {\left (28 \, c^{3} d^{2} - 43 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + {\left (4 \, c^{3} e^{2} f + 5 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )}}, -\frac {3 \, {\left (4 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - {\left (4 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\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 (2 \, c^{3} e^{2} g x^{2} - 4 \, {\left (5 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f - {\left (28 \, c^{3} d^{2} - 43 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + {\left (4 \, c^{3} e^{2} f + 5 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 623, normalized size = 2.17 \begin {gather*} \frac {\sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left ({\left ({\left (\frac {2 \, {\left (4 \, c^{4} d^{2} g e^{5} - 4 \, b c^{3} d g e^{6} + b^{2} c^{2} g e^{7}\right )} x}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}} + \frac {48 \, c^{4} d^{3} g e^{4} + 16 \, c^{4} d^{2} f e^{5} - 68 \, b c^{3} d^{2} g e^{5} - 16 \, b c^{3} d f e^{6} + 32 \, b^{2} c^{2} d g e^{6} + 4 \, b^{2} c^{2} f e^{7} - 5 \, b^{3} c g e^{7}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )} x - \frac {72 \, c^{4} d^{4} g e^{3} + 64 \, c^{4} d^{3} f e^{4} - 224 \, b c^{3} d^{3} g e^{4} - 112 \, b c^{3} d^{2} f e^{5} + 230 \, b^{2} c^{2} d^{2} g e^{5} + 64 \, b^{2} c^{2} d f e^{6} - 98 \, b^{3} c d g e^{6} - 12 \, b^{3} c f e^{7} + 15 \, b^{4} g e^{7}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )} x - \frac {112 \, c^{4} d^{5} g e^{2} + 80 \, c^{4} d^{4} f e^{3} - 284 \, b c^{3} d^{4} g e^{3} - 128 \, b c^{3} d^{3} f e^{4} + 260 \, b^{2} c^{2} d^{3} g e^{4} + 68 \, b^{2} c^{2} d^{2} f e^{5} - 103 \, b^{3} c d^{2} g e^{5} - 12 \, b^{3} c d f e^{6} + 15 \, b^{4} d g e^{6}}{4 \, c^{5} d^{2} e^{4} - 4 \, b c^{4} d e^{5} + b^{2} c^{3} e^{6}}\right )}}{4 \, {\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}} - \frac {3 \, {\left (12 \, c^{2} d^{2} g + 8 \, c^{2} d f e - 16 \, b c d g e - 4 \, b c f e^{2} + 5 \, b^{2} g e^{2}\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 )}{8 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 2032, normalized size = 7.08
result too large to display
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \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 )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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