Optimal. Leaf size=224 \[ \frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (3-2 m)-2 c (e f (3-m)-d g m)) \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (3-m) (2 c d-b e)^2 (-b e+c d-c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.33, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {794, 679, 677, 70, 69} \[ \frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (3-2 m)-2 c (e f (3-m)-d g m)) \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (3-m) (2 c d-b e)^2 (-b e+c d-c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 677
Rule 679
Rule 794
Rubi steps
\begin {align*} \int \frac {(d+e x)^m (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(b e g (3-2 m)-2 c (e f (3-m)-d g m)) \int \frac {(d+e x)^m}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{2 c e (3-m)}\\ &=\frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left ((b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^m}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx}{2 c e (3-m)}\\ &=\frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left ((b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (1+\frac {e x}{d}\right )^{\frac {1}{2}-m} \sqrt {c d^2-b d e-c d e x}\right ) \int \frac {\left (1+\frac {e x}{d}\right )^{-\frac {5}{2}+m}}{\left (c d^2-b d e-c d e x\right )^{5/2}} \, dx}{2 c e (3-m) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=\frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (c d^2 e (b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (-\frac {c d e \left (1+\frac {e x}{d}\right )}{-c d e-\frac {e \left (c d^2-b d e\right )}{d}}\right )^{\frac {1}{2}-m} \sqrt {c d^2-b d e-c d e x}\right ) \int \frac {\left (\frac {c d}{2 c d-b e}+\frac {c e x}{2 c d-b e}\right )^{-\frac {5}{2}+m}}{\left (c d^2-b d e-c d e x\right )^{5/2}} \, dx}{2 \left (-c d e-\frac {e \left (c d^2-b d e\right )}{d}\right )^2 (3-m) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}\\ &=\frac {g (d+e x)^m}{c e^2 (3-m) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(b e g (3-2 m)-2 c (e f (3-m)-d g m)) (d+e x)^m \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-m;-\frac {1}{2};\frac {c d-b e-c e x}{2 c d-b e}\right )}{3 e^2 (2 c d-b e)^2 (3-m) (c d-b e-c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 185, normalized size = 0.83 \[ \frac {2 (d+e x)^m \left (-e (d+e x) (b e-c d+c e x) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{\frac {1}{2}-m} (b e g (3-2 m)+2 c (d g m+e f (m-3))) \, _2F_1\left (-\frac {1}{2},\frac {5}{2}-m;\frac {1}{2};\frac {-c d+b e+c e x}{b e-2 c d}\right )-\frac {e (b e-2 c d)^2 (-b e g+c d g+c e f)}{c}\right )}{3 e^3 (b e-2 c d)^3 ((d+e x) (c (d-e x)-b e))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )} {\left (e x + d\right )}^{m}}{c^{3} e^{6} x^{6} + 3 \, b c^{2} e^{6} x^{5} - c^{3} d^{6} + 3 \, b c^{2} d^{5} e - 3 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 \, {\left (c^{3} d^{2} e^{4} - b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} - {\left (6 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{4} e^{2} - 2 \, b c^{2} d^{3} e^{3} + b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b c^{2} d^{4} e^{2} - 2 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (e x +d \right )^{m}}{\left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (g x + f\right )} {\left (e x + d\right )}^{m}}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^m}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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