Optimal. Leaf size=360 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}}+\frac {(2 c d-b e)^{3/2} (2 b e g-9 c d g+5 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2} \]
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Rubi [A] time = 0.61, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 6, 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 )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac {(2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt {d+e x}}+\frac {(2 c d-b e)^{3/2} (2 b e g-9 c d g+5 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} \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) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {(5 c e f-9 c d g+2 b e g) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {((-2 c d+b e) (5 c e f-9 c d g+2 b e g)) \int \frac {\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{2 e (2 c d-b e)}\\ &=-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {((2 c d-b e) (5 c e f-9 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}\\ &=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\frac {\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \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}\\ &=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}-\left ((2 c d-b e)^2 (5 c e f-9 c d g+2 b e g)\right ) \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 {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 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}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 221, normalized size = 0.61 \begin {gather*} \frac {((d+e x) (c (d-e x)-b e))^{5/2} \left ((e f-d g) (b e-c d+c e x)-\frac {(d+e x) (2 b e g-9 c d g+5 c e f) \left (\sqrt {c (d-e x)-b e} \left (23 b^2 e^2+b c e (11 e x-81 d)+c^2 \left (73 d^2-16 d e x+3 e^2 x^2\right )\right )-15 (2 c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{15 (c (d-e x)-b e)^{5/2}}\right )}{e^2 (d+e x)^{7/2} (2 c d-b e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.76, size = 382, normalized size = 1.06 \begin {gather*} \frac {\sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (46 b^2 e^2 g (d+e x)+15 b^2 d e^2 g-15 b^2 e^3 f-60 b c d^2 e g+70 b c e^2 f (d+e x)+60 b c d e^2 f-254 b c d e g (d+e x)+22 b c e g (d+e x)^2+60 c^2 d^3 g-60 c^2 d^2 e f+324 c^2 d^2 g (d+e x)-140 c^2 d e f (d+e x)+10 c^2 e f (d+e x)^2-54 c^2 d g (d+e x)^2+6 c^2 g (d+e x)^3\right )}{15 e^2 (d+e x)^{3/2}}+\frac {\left (-2 b^3 e^3 g+17 b^2 c d e^2 g-5 b^2 c e^3 f-44 b c^2 d^2 e g+20 b c^2 d e^2 f+36 c^3 d^3 g-20 c^3 d^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 )}{e^2 \sqrt {b e-2 c d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 990, normalized size = 2.75 \begin {gather*} \left [-\frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 \, {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{30 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 ) + {\left (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1136, normalized size = 3.16 \begin {gather*} -\frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (30 b^{3} e^{4} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-255 b^{2} c d \,e^{3} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+75 b^{2} c \,e^{4} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 b \,c^{2} d^{2} e^{2} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-300 b \,c^{2} d \,e^{3} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-540 c^{3} d^{3} e g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+300 c^{3} d^{2} e^{2} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+30 b^{3} d \,e^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-255 b^{2} c \,d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+75 b^{2} c d \,e^{3} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+660 b \,c^{2} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-300 b \,c^{2} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-540 c^{3} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+300 c^{3} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} g \,x^{3}-22 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}+36 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-10 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}-46 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} g x +210 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} g x -70 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{3} f x -234 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e g x +120 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x -61 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +15 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{3} f +292 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,d^{2} e g -130 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d \,e^{2} f -336 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{3} g +190 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{15 \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 {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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