Optimal. Leaf size=278 \[ \frac {(2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16 c^{3/2} e^2}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \]
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Rubi [A] time = 0.42, antiderivative size = 278, 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 = {792, 664, 612, 621, 204} \begin {gather*} \frac {(2 c d-b e)^2 (-b e g-4 c d g+6 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 )}{16 c^{3/2} e^2}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-4 c d g+6 c e f)}{3 e^2 (2 c d-b e)}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-4 c d g+6 c e f)}{8 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 612
Rule 621
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx &=\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac {(6 c e f-4 c d g-b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{d+e x} \, dx}{e (2 c d-b e)}\\ &=\frac {(6 c e f-4 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}-\frac {((-2 c d+b e) (6 c e f-4 c d g-b e g)) \int \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{2 e (2 c d-b e)}\\ &=\frac {(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac {(6 c e f-4 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac {\left ((2 c d-b e)^2 (6 c e f-4 c d g-b e g)\right ) \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{16 c e}\\ &=\frac {(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac {(6 c e f-4 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac {\left ((2 c d-b e)^2 (6 c e f-4 c d g-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 )}{8 c e}\\ &=\frac {(6 c e f-4 c d g-b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{8 c e}+\frac {(6 c e f-4 c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^2}+\frac {(2 c d-b e)^2 (6 c e f-4 c d g-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 )}{16 c^{3/2} e^2}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 228, normalized size = 0.82 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (-\sqrt {c} \sqrt {e} \left (3 b^2 e^2 g+2 b c e (-14 d g+15 e f+7 e g x)+4 c^2 \left (10 d^2 g-6 d e (2 f+g x)+e^2 x (3 f+2 g x)\right )\right )-\frac {3 \sqrt {e (2 c d-b e)} (b e-2 c d) (-b e g-4 c d g+6 c e f) \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )}{\sqrt {d+e x} \sqrt {\frac {b e-c d+c e x}{b e-2 c d}}}\right )}{24 c^{3/2} e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 23.18, size = 21808, normalized size = 78.45 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 567, normalized size = 2.04 \begin {gather*} \left [\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f - {\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\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 (8 \, c^{3} e^{2} g x^{2} - 6 \, {\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f - {\left (12 \, c^{3} d e - 7 \, 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}}{96 \, c^{2} e^{2}}, -\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f - {\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\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 (8 \, c^{3} e^{2} g x^{2} - 6 \, {\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f - {\left (12 \, c^{3} d e - 7 \, 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}}{48 \, c^{2} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 2174, normalized size = 7.82
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 (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^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 (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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