Optimal. Leaf size=129 \[ \frac {2 (d+e x) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g \tan ^{-1}\left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e^2} \]
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Rubi [A] time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {777, 621, 204} \begin {gather*} \frac {2 (e x (2 c d-b e)+d (2 c d-b e)) (-b e g+c d g+c e f)}{c e^2 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {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 )}{c^{3/2} e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 777
Rubi steps
\begin {align*} \int \frac {(d+e x) (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)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {g \int \frac {1}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(2 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 )}{c e}\\ &=\frac {2 (c e f+c d g-b e g) (d+e x)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {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 )}{c^{3/2} e^2}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 173, normalized size = 1.34 \begin {gather*} -\frac {2 \left (\sqrt {c} \sqrt {e} (d+e x) (-b e g+c d g+c e f)+g \sqrt {d+e x} \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}} \sin ^{-1}\left (\frac {\sqrt {c} \sqrt {e} \sqrt {d+e x}}{\sqrt {e (2 c d-b e)}}\right )\right )}{c^{3/2} e^{5/2} (b e-2 c d) \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 1.70, size = 267, normalized size = 2.07 \begin {gather*} -\frac {g \sqrt {-c e^2} \log \left (b^2 e^2-8 c x \sqrt {-c e^2} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}-4 b c d e-4 b c e^2 x+4 c^2 d^2-8 c^2 e^2 x^2\right )}{2 c^2 e^3}-\frac {g \tan ^{-1}\left (\frac {2 \sqrt {c} x \sqrt {-c e^2}}{b e}-\frac {2 \sqrt {c} \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{b e}\right )}{c^{3/2} e^2}+\frac {2 \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} (-b e g+c d g+c e f)}{c e^2 (b e-2 c d) (b e-c d+c e x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 483, normalized size = 3.74 \begin {gather*} \left [\frac {{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x - {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\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 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c^{2} e f + {\left (c^{2} d - b c e\right )} g\right )}}{2 \, {\left (2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} - {\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x\right )}}, -\frac {{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} g x - {\left (2 \, c^{2} d^{2} - 3 \, b c d e + b^{2} e^{2}\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 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c^{2} e f + {\left (c^{2} d - b c e\right )} g\right )}}{2 \, c^{4} d^{2} e^{2} - 3 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4} - {\left (2 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 282, normalized size = 2.19 \begin {gather*} -\frac {\sqrt {-c e^{2}} g 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 )}{c^{2}} - \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e} {\left (\frac {{\left (2 \, c^{2} d^{2} g e^{2} + 2 \, c^{2} d f e^{3} - 3 \, b c d g e^{3} - b c f e^{4} + b^{2} g e^{4}\right )} x}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}} + \frac {2 \, c^{2} d^{3} g e + 2 \, c^{2} d^{2} f e^{2} - 3 \, b c d^{2} g e^{2} - b c d f e^{3} + b^{2} d g e^{3}}{4 \, c^{3} d^{2} e^{3} - 4 \, b c^{2} d e^{4} + b^{2} c e^{5}}\right )}}{c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 710, normalized size = 5.50 \begin {gather*} -\frac {b^{2} e^{3} g x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {2 b d \,e^{2} g x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}+\frac {2 b \,e^{3} f x}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}-\frac {b^{3} e^{3} g}{2 \left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2}}+\frac {b^{2} d \,e^{2} g}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {b^{2} e^{3} f}{\left (-b^{2} e^{4}+4 b c d \,e^{3}-4 d^{2} e^{2} c^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c}+\frac {2 \left (-2 c \,e^{2} x -b \,e^{2}\right ) d f}{\left (-b^{2} e^{4}-4 \left (-b d e +c \,d^{2}\right ) c \,e^{2}\right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}+\frac {g x}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c e}-\frac {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 e}-\frac {b g}{2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{2} e}+\frac {d g}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c \,e^{2}}+\frac {f}{\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c e} \end {gather*}
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 [B] time = 4.04, size = 344, normalized size = 2.67 \begin {gather*} \frac {4\,c\,d^3\,g+2\,b\,d\,e^2\,f-4\,b\,d^2\,e\,g-2\,b\,d\,e^2\,g\,x+4\,c\,d\,e^2\,f\,x}{\left (b^2\,e^4+4\,c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}+\frac {e\,g\,\ln \left (b\,e^2-2\,\sqrt {-c\,e^2}\,\sqrt {-\left (d+e\,x\right )\,\left (b\,e-c\,d+c\,e\,x\right )}+2\,c\,e^2\,x\right )}{{\left (-c\,e^2\right )}^{3/2}}-\frac {e\,f\,\left (-4\,c\,d^2+4\,b\,d\,e+2\,b\,x\,e^2\right )}{\left (b^2\,e^4+4\,c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}+\frac {g\,\left (x\,\left (\frac {b^2\,e^4}{2}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )-\frac {b\,e^2\,\left (c\,d^2-b\,d\,e\right )}{2}\right )}{c\,e\,\left (\frac {b^2\,e^4}{4}+c\,e^2\,\left (c\,d^2-b\,d\,e\right )\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,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 ) \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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