Optimal. Leaf size=250 \[ \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e)^{3/2} (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}} \]
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Rubi [A] time = 0.45, antiderivative size = 250, 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 = {794, 664, 660, 208} \begin {gather*} \frac {2 (e f-d 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 {2 (2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e)^{3/2} (e f-d g) \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}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 664
Rule 794
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)^{5/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}}-\frac {\left (2 \left (\frac {5}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \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}{5 c e^3}\\ &=\frac {2 (e f-d 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 {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}}+\frac {((2 c d-b e) (e f-d 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}{e}\\ &=\frac {2 (2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (e f-d 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 {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}}+\frac {\left ((2 c d-b e)^2 (e f-d 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}{e}\\ &=\frac {2 (2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (e f-d 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 {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}}+\left (2 (2 c d-b e)^2 (e f-d 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 (2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 (e f-d 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 {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}}-\frac {2 (2 c d-b e)^{3/2} (e f-d 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 = 0.28, size = 199, normalized size = 0.80 \begin {gather*} -\frac {2 \sqrt {d+e x} \sqrt {c (d-e x)-b e} \left (\sqrt {c (d-e x)-b e} \left (3 b^2 e^2 g+2 b c e (-13 d g+10 e f+3 e g x)+c^2 \left (38 d^2 g-d e (35 f+11 g x)+e^2 x (5 f+3 g x)\right )\right )-15 c (2 c d-b e)^{3/2} (d g-e f) \tanh ^{-1}\left (\frac {\sqrt {-b e+c d-c e x}}{\sqrt {2 c d-b e}}\right )\right )}{15 c e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.13, size = 238, normalized size = 0.95 \begin {gather*} -\frac {2 \sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (3 b^2 e^2 g+6 b c e g (d+e x)-32 b c d e g+20 b c e^2 f+52 c^2 d^2 g+5 c^2 e f (d+e x)-40 c^2 d e f+3 c^2 g (d+e x)^2-17 c^2 d g (d+e x)\right )}{15 c e^2 \sqrt {d+e x}}-\frac {2 (b e-2 c d)^{3/2} (d g-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} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 656, normalized size = 2.62 \begin {gather*} \left [\frac {15 \, \sqrt {2 \, c d - b e} {\left ({\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (2 \, c^{2} d^{3} - b c d^{2} e\right )} g + {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} g\right )} x\right )} \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 (3 \, c^{2} e^{2} g x^{2} - 5 \, {\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} f + {\left (38 \, c^{2} d^{2} - 26 \, b c d e + 3 \, b^{2} e^{2}\right )} g + {\left (5 \, c^{2} e^{2} f - {\left (11 \, c^{2} d e - 6 \, b c 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}}{15 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (15 \, \sqrt {-2 \, c d + b e} {\left ({\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (2 \, c^{2} d^{3} - b c d^{2} e\right )} g + {\left ({\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} g\right )} x\right )} \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 (3 \, c^{2} e^{2} g x^{2} - 5 \, {\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} f + {\left (38 \, c^{2} d^{2} - 26 \, b c d e + 3 \, b^{2} e^{2}\right )} g + {\left (5 \, c^{2} e^{2} f - {\left (11 \, c^{2} d e - 6 \, b c 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}\right )}}{15 \, {\left (c e^{3} x + c d e^{2}\right )}}\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: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 601, normalized size = 2.40 \begin {gather*} -\frac {2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (15 b^{2} c d \,e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-15 b^{2} c \,e^{3} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-60 b \,c^{2} d^{2} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+60 b \,c^{2} d \,e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+60 c^{3} d^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-60 c^{3} d^{2} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{2} g \,x^{2}+6 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{2} g x -11 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d e g x +5 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} e^{2} f x +3 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} e^{2} g -26 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c d e g +20 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b c \,e^{2} f +38 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d^{2} g -35 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{2} d e f \right )}{15 \sqrt {e x +d}\, \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c \,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 {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 )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\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 {\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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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