Optimal. Leaf size=572 \[ -\frac {(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac {(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac {(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b (c f (m+2)-d e (m+3)))}{d^3 f (m+3) (d e-c f)}-\frac {h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac {h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f-b (2 d e (m+2)-c f (2 m+3)))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac {b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m} \]
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Rubi [A] time = 0.66, antiderivative size = 566, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {159, 89, 79, 70, 69, 90, 45, 37} \[ -\frac {(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac {(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac {(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d^3 f (m+3) (d e-c f)}-\frac {b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac {h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac {h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 69
Rule 70
Rule 79
Rule 89
Rule 90
Rule 159
Rubi steps
\begin {align*} \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx &=\frac {h \int (a+b x)^2 (c+d x)^{-3-m} (e+f x)^m \, dx}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m \, dx}{d}\\ &=-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \int (c+d x)^{-4-m} (e+f x)^m \left (-a^2 d f-b (b c e+a c f (1+m)-a d e (3+m))+b^2 (d e-c f) (2+m) x\right ) \, dx}{d^2 f}+\frac {h \int (c+d x)^{-2-m} (e+f x)^m \left (-a^2 d^2 f+b^2 c (c f (1+m)-d e (2+m))-2 a b d (c f (1+m)-d e (2+m))+b^2 d (d e-c f) (2+m) x\right ) \, dx}{d^3 (d e-c f) (2+m)}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}+\frac {\left (b^2 h\right ) \int (c+d x)^{-1-m} (e+f x)^m \, dx}{d^3}+\frac {\left ((d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right )\right ) \int (c+d x)^{-3-m} (e+f x)^m \, dx}{d^2 f (3+m)}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^2 f (d e-c f) (2+m) (3+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}-\frac {\left ((d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right )\right ) \int (c+d x)^{-2-m} (e+f x)^m \, dx}{d^2 (d e-c f) (2+m) (3+m)}+\frac {\left (b^2 h (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m}\right ) \int (c+d x)^{-1-m} \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^m \, dx}{d^3}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^2 f (d e-c f) (2+m) (3+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}+\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^2 (d e-c f)^2 (1+m) (2+m) (3+m)}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m}\\ \end {align*}
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Mathematica [A] time = 1.91, size = 422, normalized size = 0.74 \[ \frac {(c+d x)^{-m-3} (e+f x)^m \left (-h (m+3) (c+d x) (d e-c f) \left (d f (m+1) (e+f x) (b c-a d)^2 (d e-c f)-(c+d x) \left (d (e+f x) \left (a^2 d^2 f^2+2 a b d f (c f (m+1)-d e (m+2))+b^2 \left (d^2 e^2 (m+2)-c^2 f^2 (m+1)\right )\right )-b^2 (m+2) (d e-c f)^3 \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m-1,-m-1;-m;\frac {f (c+d x)}{c f-d e}\right )\right )\right )-d (e+f x) (d g-c h) \left ((c+d x) (d (e m+e-f x)-c f (m+2)) \left (2 d f \left (a^2 (-d) f-b (a c f (m+1)-a d e (m+3)+b c e)\right )+b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))\right )+b d (m+1) (m+2) (m+3) (a+b x) (d e-c f)^3-(m+1) (m+2) (b c-a d) (d e-c f)^2 (a d f+b c f (m+2)-b d e (m+3))\right )\right )}{d^4 f (m+1) (m+2) (m+3) (d e-c f)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} h x^{3} + a^{2} g + {\left (b^{2} g + 2 \, a b h\right )} x^{2} + {\left (2 \, a b g + a^{2} h\right )} x\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} \left (h x +g \right ) \left (d x +c \right )^{-m -4} \left (f x +e \right )^{m}\, dx \]
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 \[ \int {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^m\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{m+4}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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