Optimal. Leaf size=288 \[ -\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-7 c d g+3 c e f)}{3 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-7 c d g+3 c e f)}{e^2 \sqrt {d+e x}}+\frac {\sqrt {2 c d-b e} (2 b e g-7 c d g+3 c e f) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2} \]
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Rubi [A] time = 0.48, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 5, 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 )^{5/2}}{e^2 (d+e x)^{7/2} (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-7 c d g+3 c e f)}{3 e^2 (d+e x)^{3/2} (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-7 c d g+3 c e f)}{e^2 \sqrt {d+e x}}+\frac {\sqrt {2 c d-b e} (2 b e g-7 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 )}{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 )^{3/2}}{(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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {(3 c e f-7 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 {(3 c e f-7 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 (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {((-2 c d+b e) (3 c e f-7 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 {(3 c e f-7 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 {(3 c e f-7 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 (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-\frac {((2 c d-b e) (3 c e f-7 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}\\ &=-\frac {(3 c e f-7 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 {(3 c e f-7 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 (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}-((2 c d-b e) (3 c e f-7 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 {(3 c e f-7 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 {(3 c e f-7 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 (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 )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {\sqrt {2 c d-b e} (3 c e f-7 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 = 0.26, size = 195, normalized size = 0.68 \begin {gather*} \frac {\sqrt {(d+e x) (c (d-e x)-b e)} \left (\sqrt {c (d-e x)-b e} \left (b e (-11 d g+3 e f-8 e g x)+2 c \left (13 d^2 g+d e (9 g x-6 f)-e^2 x (3 f+g x)\right )\right )-3 (d+e x) \sqrt {2 c d-b e} (-2 b e g+7 c d g-3 c e f) \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{3 e^2 (d+e x)^{3/2} \sqrt {c (d-e x)-b e}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.32, size = 246, normalized size = 0.85 \begin {gather*} \frac {\left (2 b^2 e^2 g-11 b c d e g+3 b c e^2 f+14 c^2 d^2 g-6 c^2 d 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}}+\frac {\sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (-8 b e g (d+e x)-3 b d e g+3 b e^2 f+6 c d^2 g-6 c e f (d+e x)-6 c d e f+22 c d g (d+e x)-2 c g (d+e x)^2\right )}{3 e^2 (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 639, normalized size = 2.22 \begin {gather*} \left [\frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, 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 \, {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\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}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, 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 ) - {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\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}}{3 \, {\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 = 695, normalized size = 2.41 \begin {gather*} \frac {\sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (6 b^{2} e^{3} g x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-33 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 )+9 b c \,e^{3} f x \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+42 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 )-18 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 )+6 b^{2} d \,e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-33 b c \,d^{2} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+9 b c d \,e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+42 c^{2} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-18 c^{2} d^{2} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-2 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2} g \,x^{2}-8 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} g x +18 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e g x -6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,e^{2} f x -11 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b d e g +3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,e^{2} f +26 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,d^{2} g -12 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c d e f \right )}{3 \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 {3}{2}} {\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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\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 {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \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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