Optimal. Leaf size=233 \[ -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac {c (-4 b e g+5 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{5/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 233, 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, 672, 660, 208} \begin {gather*} -\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{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} (-4 b e g+5 c d g+3 c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac {c (-4 b e g+5 c d g+3 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 )}{4 e^2 (2 c d-b e)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {f+g x}{(d+e x)^{5/2} \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}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}+\frac {(3 c e f+5 c d g-4 b e g) \int \frac {1}{(d+e x)^{3/2} \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{4 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}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(3 c e f+5 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac {(c (3 c e f+5 c d g-4 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}{8 e (2 c d-b e)^2}\\ &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(3 c e f+5 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}+\frac {(c (3 c e f+5 c d g-4 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 )}{4 (2 c d-b e)^2}\\ &=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(3 c e f+5 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e)^2 (d+e x)^{3/2}}-\frac {c (3 c e f+5 c d g-4 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 )}{4 e^2 (2 c d-b e)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 209, normalized size = 0.90 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (-\frac {(d+e x) (-4 b e g+5 c d g+3 c e f) \left (\sqrt {e} (2 c d-b e) \sqrt {c (d-e x)-b e}+c (d+e x) \sqrt {e (b e-2 c d)} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c (d-e x)-b e}}{\sqrt {e (b e-2 c d)}}\right )\right )}{2 e^{3/2} (b e-2 c d)^2 \sqrt {c (d-e x)-b e}}+\frac {d g}{e}-f\right )}{2 e (d+e x)^{5/2} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.96, size = 238, normalized size = 1.02 \begin {gather*} \frac {\left (-4 b c e g+5 c^2 d g+3 c^2 e f\right ) \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 )}{4 e^2 (2 c d-b e)^2 \sqrt {b e-2 c d}}+\frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (4 b e g (d+e x)-2 b d e g+2 b e^2 f+4 c d^2 g-3 c e f (d+e x)-4 c d e f-5 c d g (d+e x)\right )}{4 e^2 (d+e x)^{5/2} (b e-2 c d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 1168, normalized size = 5.01
result too large to display
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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 630, normalized size = 2.70 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (4 b c \,e^{3} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-5 c^{2} d \,e^{2} g \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c^{2} e^{3} f \,x^{2} \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+8 b c d \,e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-10 c^{2} d^{2} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-6 c^{2} d \,e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+4 b c \,d^{2} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-5 c^{2} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-3 c^{2} d^{2} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-4 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} g x +5 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e g x +3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2} f x -2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b d e g -2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} f +\sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,d^{2} g +7 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e f \right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (b e -2 c d \right )^{\frac {5}{2}} \sqrt {-c e x -b e +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 {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 {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 {f+g\,x}{{\left (d+e\,x\right )}^{5/2}\,\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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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