Optimal. Leaf size=223 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2 \sqrt {2 c d-b e}} \]
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Rubi [A] time = 0.36, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {792, 664, 660, 208} \begin {gather*} -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt {d+e x} (2 c d-b e)}+\frac {(2 b e g-5 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 \sqrt {2 c d-b e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(c e f-5 c d g+2 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) \sqrt {d+e x}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {((-2 c d+b e) (c e f-5 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 {(c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) \sqrt {d+e x}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^{5/2}}+(-c e f+5 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 )\\ &=-\frac {(c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (2 c d-b e) \sqrt {d+e x}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(c e f-5 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 \sqrt {2 c d-b e}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 173, normalized size = 0.78 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left ((d+e x) \sqrt {2 c d-b e} (-2 b e g+5 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 )-(2 c d-b e) \sqrt {c (d-e x)-b e} (3 d g-e f+2 e g x)\right )}{e^2 (d+e x)^{3/2} (b e-2 c d) \sqrt {c (d-e x)-b e}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.87, size = 165, normalized size = 0.74 \begin {gather*} \frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} (2 g (d+e x)+d g-e f)}{e^2 (d+e x)^{3/2}}+\frac {(-2 b e g+5 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 \sqrt {b e-2 c d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 659, normalized size = 2.96 \begin {gather*} \left [\frac {{\left (c d^{2} e f + {\left (c e^{3} f - {\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f - {\left (5 \, 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 (2 \, {\left (2 \, c d e - b e^{2}\right )} g x - {\left (2 \, c d e - b e^{2}\right )} f + 3 \, {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{2 \, {\left (2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x\right )}}, \frac {{\left (c d^{2} e f + {\left (c e^{3} f - {\left (5 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (5 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (c d e^{2} f - {\left (5 \, 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 (2 \, {\left (2 \, c d e - b e^{2}\right )} g x - {\left (2 \, c d e - b e^{2}\right )} f + 3 \, {\left (2 \, c d^{2} - b d e\right )} g\right )} \sqrt {e x + d}}{2 \, c d^{3} e^{2} - b d^{2} e^{3} + {\left (2 \, c d e^{4} - b e^{5}\right )} x^{2} + 2 \, {\left (2 \, c d^{2} e^{3} - b d e^{4}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 359, normalized size = 1.61 \begin {gather*} \frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \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 )+5 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 )+5 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 )+2 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, e g x +3 \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 )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, 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 {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {5}{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 = -0.00 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{5/2}} \,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 {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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