Optimal. Leaf size=231 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {c (-4 b e g+7 c d g+c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 231, 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, 662, 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}}{2 e^2 (d+e x)^{7/2} (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+7 c d g+c e f)}{4 e^2 (d+e x)^{3/2} (2 c d-b e)}+\frac {c (-4 b e g+7 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 )}{4 e^2 (2 c d-b e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 662
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)^{7/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {(c e f+7 c d g-4 b e g) \int \frac {\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx}{4 e (2 c d-b e)}\\ &=-\frac {(c e f+7 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) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 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)}\\ &=-\frac {(c e f+7 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) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(c (c e f+7 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)}\\ &=-\frac {(c e f+7 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) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {c (c e f+7 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)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 222, normalized size = 0.96 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (-2 e (e f-d g) (b e-c d+c e x)^2-\frac {e (d+e x) (-4 b e g+7 c d g+c e f) \left (c \sqrt {e} (d+e x) \sqrt {c (d-e x)-b e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c (d-e x)-b e}}{\sqrt {e (b e-2 c d)}}\right )-\sqrt {e (b e-2 c d)} (b e-c d+c e x)\right )}{\sqrt {e (b e-2 c d)}}\right )}{4 e^3 (d+e x)^{5/2} (b e-2 c d) (b e-c d+c e x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.18, size = 238, normalized size = 1.03 \begin {gather*} \frac {\left (4 b c e g-7 c^2 d g+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) \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-c e f (d+e x)+4 c d e f+9 c d g (d+e x)\right )}{4 e^2 (d+e x)^{5/2} (b e-2 c d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 1043, normalized size = 4.52 \begin {gather*} \left [\frac {{\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f + {\left (7 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f + {\left (7 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} + {\left (7 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f + {\left (7 \, c^{2} d^{3} e - 4 \, b c d^{2} 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 (6 \, c^{2} d^{2} e - 7 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} f + {\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (18 \, c^{2} d^{2} e - 17 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}}, \frac {{\left (c^{2} d^{3} e f + {\left (c^{2} e^{4} f + {\left (7 \, c^{2} d e^{3} - 4 \, b c e^{4}\right )} g\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} f + {\left (7 \, c^{2} d^{2} e^{2} - 4 \, b c d e^{3}\right )} g\right )} x^{2} + {\left (7 \, c^{2} d^{4} - 4 \, b c d^{3} e\right )} g + 3 \, {\left (c^{2} d^{2} e^{2} f + {\left (7 \, c^{2} d^{3} e - 4 \, b c d^{2} 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 (6 \, c^{2} d^{2} e - 7 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} f + {\left (10 \, c^{2} d^{3} - 9 \, b c d^{2} e + 2 \, b^{2} d e^{2}\right )} g - {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (18 \, c^{2} d^{2} e - 17 \, b c d e^{2} + 4 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (4 \, c^{2} d^{5} e^{2} - 4 \, b c d^{4} e^{3} + b^{2} d^{3} e^{4} + {\left (4 \, c^{2} d^{2} e^{5} - 4 \, b c d e^{6} + b^{2} e^{7}\right )} x^{3} + 3 \, {\left (4 \, c^{2} d^{3} e^{4} - 4 \, b c d^{2} e^{5} + b^{2} d e^{6}\right )} x^{2} + 3 \, {\left (4 \, c^{2} d^{4} e^{3} - 4 \, b c d^{3} e^{4} + b^{2} d^{2} e^{5}\right )} x\right )}}\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 {\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 {7}{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 = 630, normalized size = 2.73 \begin {gather*} -\frac {\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 )-7 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 )-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 )-14 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 )-2 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 )-7 c^{2} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-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 -9 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e g x +\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 -5 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,d^{2} g -3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{4 \left (b e -2 c d \right )^{\frac {3}{2}} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {5}{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 {\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 {7}{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 )}^{7/2}} \,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 {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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