Optimal. Leaf size=316 \[ \frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e)^{5/2} (e f-d g) \tanh ^{-1}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e^2}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \]
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Rubi [A] time = 0.59, antiderivative size = 316, 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 = {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 )^{5/2}}{5 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (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)^2 (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)^{5/2} (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 )^{7/2}}{7 c e^2 (d+e x)^{7/2}} \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 )^{5/2}}{(d+e x)^{7/2}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )-\frac {7}{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 )^{5/2}}{(d+e x)^{7/2}} \, dx}{7 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 )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {((2 c d-b e) (e f-d 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}{e}\\ &=\frac {2 (2 c d-b e) (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 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^2 (e f-d g)\right ) \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)^2 (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) (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 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\frac {\left ((2 c d-b e)^3 (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)^2 (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) (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 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}+\left (2 (2 c d-b e)^3 (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)^2 (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) (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 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c e^2 (d+e x)^{7/2}}-\frac {2 (2 c d-b e)^{5/2} (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 = 0.57, size = 197, normalized size = 0.62 \begin {gather*} \frac {2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac {7 c (e f-d g) \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}}+g (b e-c d+c e x)\right )}{7 c e^2 (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.62, size = 347, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {-b e (d+e x)-c (d+e x)^2+2 c d (d+e x)} \left (15 b^3 e^3 g+45 b^2 c e^2 g (d+e x)-251 b^2 c d e^2 g+161 b^2 c e^3 f+824 b c^2 d^2 e g+77 b c^2 e^2 f (d+e x)-644 b c^2 d e^2 f+45 b c^2 e g (d+e x)^2-257 b c^2 d e g (d+e x)-764 c^3 d^3 g+644 c^3 d^2 e f+334 c^3 d^2 g (d+e x)+21 c^3 e f (d+e x)^2-154 c^3 d e f (d+e x)+15 c^3 g (d+e x)^3-111 c^3 d g (d+e x)^2\right )}{105 c e^2 \sqrt {d+e x}}+\frac {2 (b e-2 c d)^{5/2} (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 = 0.43, size = 949, normalized size = 3.00 \begin {gather*} \left [-\frac {105 \, \sqrt {2 \, c d - b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\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 (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c 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}}{105 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (105 \, \sqrt {-2 \, c d + b e} {\left ({\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\right )} f - {\left (4 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} g + {\left ({\left (4 \, c^{3} d^{2} e^{2} - 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} f - {\left (4 \, c^{3} d^{3} e - 4 \, b c^{2} d^{2} e^{2} + b^{2} c d e^{3}\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 (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f - {\left (22 \, c^{3} d e^{2} - 15 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (73 \, c^{3} d^{2} e - 81 \, b c^{2} d e^{2} + 23 \, b^{2} c e^{3}\right )} f - {\left (526 \, c^{3} d^{3} - 612 \, b c^{2} d^{2} e + 206 \, b^{2} c d e^{2} - 15 \, b^{3} e^{3}\right )} g - {\left (7 \, {\left (16 \, c^{3} d e^{2} - 11 \, b c^{2} e^{3}\right )} f - {\left (157 \, c^{3} d^{2} e - 167 \, b c^{2} d e^{2} + 45 \, b^{2} c 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}\right )}}{105 \, {\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(-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 = 0.07, size = 956, normalized size = 3.03 \begin {gather*} \frac {2 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, \left (105 b^{3} c d \,e^{3} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-105 b^{3} c \,e^{4} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-630 b^{2} c^{2} d^{2} e^{2} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+630 b^{2} c^{2} d \,e^{3} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+1260 b \,c^{3} d^{3} e g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-1260 b \,c^{3} d^{2} e^{2} f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )-840 c^{4} d^{4} g \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+840 c^{4} d^{3} e f \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right )+15 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} e^{3} g \,x^{3}+45 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} e^{3} g \,x^{2}-66 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d \,e^{2} g \,x^{2}+21 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} e^{3} f \,x^{2}+45 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b^{2} c \,e^{3} g x -167 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} d \,e^{2} g x +77 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, b \,c^{2} e^{3} f x +157 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d^{2} e g x -112 \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\, c^{3} d \,e^{2} f x +15 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{3} e^{3} g -206 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{2} c d \,e^{2} g +161 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b^{2} c \,e^{3} f +612 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d^{2} e g -567 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, b \,c^{2} d \,e^{2} f -526 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, c^{3} d^{3} g +511 \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}\, c^{3} d^{2} e f \right )}{105 \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 {5}{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 )}^{5/2}}{{\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(-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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