Optimal. Leaf size=153 \[ -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac {(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {792, 660, 208} \begin {gather*} -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac {(-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 (2 c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 792
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(c e f+3 c d g-2 b e g) \int \frac {1}{\sqrt {d+e x} \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 {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {(c e f+3 c d g-2 b e g) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+b e^3+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{\sqrt {d+e x}}\right )}{2 c d-b e}\\ &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(c e f+3 c d g-2 b e g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2 (2 c d-b e)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 153, normalized size = 1.00 \begin {gather*} \frac {(e f-d g) (b e-c d+c e x)-\frac {(d+e x) \sqrt {c (d-e x)-b e} (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )}{\sqrt {2 c d-b e}}}{e^2 \sqrt {d+e x} (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 [A] time = 0.73, size = 167, normalized size = 1.09 \begin {gather*} \frac {(e f-d g) \sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)}}{e^2 (d+e x)^{3/2} (b e-2 c d)}+\frac {(2 b e g-3 c d g-c e f) \tan ^{-1}\left (\frac {\sqrt {b e-2 c d} \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2}}{\sqrt {d+e x} (b e+c (d+e x)-2 c d)}\right )}{e^2 (b e-2 c d)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 698, normalized size = 4.56 \begin {gather*} \left [\frac {{\left (c d^{2} e f + {\left (c e^{3} f + {\left (3 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} + {\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f + {\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x + 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c d e - b e^{2}\right )} f - {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{2 \, {\left (4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} + {\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \, {\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x\right )}}, -\frac {{\left (c d^{2} e f + {\left (c e^{3} f + {\left (3 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} + {\left (3 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f + {\left (3 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) + \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left ({\left (2 \, c d e - b e^{2}\right )} f - {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{4 \, c^{2} d^{4} e^{2} - 4 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4} + {\left (4 \, c^{2} d^{2} e^{4} - 4 \, b c d e^{5} + b^{2} e^{6}\right )} x^{2} + 2 \, {\left (4 \, c^{2} d^{3} e^{3} - 4 \, b c d^{2} e^{4} + b^{2} d e^{5}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {g x + f}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 328, normalized size = 2.14 \begin {gather*} \frac {\left (2 b \,e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c d e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-c \,e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+2 b d e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c \,d^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-c d e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, d g +\sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{\left (b e -2 c d \right )^{\frac {3}{2}} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {3}{2}} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {g x + f}{\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \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 = -0.01 \begin {gather*} \int \frac {f+g\,x}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}} \,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 {f + g x}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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