Optimal. Leaf size=343 \[ -\frac {6 (d+e x)^{m-1} (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac {6 g (d+e x)^m (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac {3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac {(f+g x)^3 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (4-m)} \]
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Rubi [A] time = 0.45, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {870, 794, 648} \begin {gather*} -\frac {6 (d+e x)^{m-1} (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac {6 g (d+e x)^m (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac {3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac {(f+g x)^3 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (4-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 794
Rule 870
Rubi steps
\begin {align*} \int (d+e x)^m (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx &=\frac {(d+e x)^{-1+m} (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (4-m)}+\frac {(3 (c d f-a e g)) \int (d+e x)^m (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c d (4-m)}\\ &=\frac {3 (c d f-a e g) (d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac {(d+e x)^{-1+m} (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (4-m)}+\frac {\left (6 (c d f-a e g)^2\right ) \int (d+e x)^m (f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c^2 d^2 (3-m) (4-m)}\\ &=\frac {6 g (c d f-a e g)^2 (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac {3 (c d f-a e g) (d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac {(d+e x)^{-1+m} (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (4-m)}-\frac {\left (6 (c d f-a e g)^2 \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )\right ) \int (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx}{c^3 d^3 e (2-m) (3-m) (4-m)}\\ &=-\frac {6 (c d f-a e g)^2 \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right ) (d+e x)^{-1+m} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac {6 g (c d f-a e g)^2 (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac {3 (c d f-a e g) (d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac {(d+e x)^{-1+m} (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (4-m)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 134, normalized size = 0.39 \begin {gather*} \frac {(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left (\frac {3 g^2 (a e+c d x)^2 (a e g-c d f)}{m-3}-\frac {3 g (a e+c d x) (c d f-a e g)^2}{m-2}-\frac {(c d f-a e g)^3}{m-1}-\frac {g^3 (a e+c d x)^3}{m-4}\right )}{c^4 d^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.44, size = 705, normalized size = 2.06 \begin {gather*} -\frac {{\left (a c^{3} d^{3} e f^{3} m^{3} - 24 \, a c^{3} d^{3} e f^{3} + 36 \, a^{2} c^{2} d^{2} e^{2} f^{2} g - 24 \, a^{3} c d e^{3} f g^{2} + 6 \, a^{4} e^{4} g^{3} + {\left (c^{4} d^{4} g^{3} m^{3} - 6 \, c^{4} d^{4} g^{3} m^{2} + 11 \, c^{4} d^{4} g^{3} m - 6 \, c^{4} d^{4} g^{3}\right )} x^{4} - {\left (24 \, c^{4} d^{4} f g^{2} - {\left (3 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{3} + 3 \, {\left (7 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{2} - 2 \, {\left (21 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m\right )} x^{3} - 3 \, {\left (3 \, a c^{3} d^{3} e f^{3} - a^{2} c^{2} d^{2} e^{2} f^{2} g\right )} m^{2} - 3 \, {\left (12 \, c^{4} d^{4} f^{2} g - {\left (c^{4} d^{4} f^{2} g + a c^{3} d^{3} e f g^{2}\right )} m^{3} + {\left (8 \, c^{4} d^{4} f^{2} g + 5 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m^{2} - {\left (19 \, c^{4} d^{4} f^{2} g + 4 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m\right )} x^{2} + {\left (26 \, a c^{3} d^{3} e f^{3} - 21 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 6 \, a^{3} c d e^{3} f g^{2}\right )} m - {\left (24 \, c^{4} d^{4} f^{3} - {\left (c^{4} d^{4} f^{3} + 3 \, a c^{3} d^{3} e f^{2} g\right )} m^{3} + 3 \, {\left (3 \, c^{4} d^{4} f^{3} + 7 \, a c^{3} d^{3} e f^{2} g - 2 \, a^{2} c^{2} d^{2} e^{2} f g^{2}\right )} m^{2} - 2 \, {\left (13 \, c^{4} d^{4} f^{3} + 18 \, a c^{3} d^{3} e f^{2} g - 12 \, a^{2} c^{2} d^{2} e^{2} f g^{2} + 3 \, a^{3} c d e^{3} g^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{{\left (c^{4} d^{4} m^{4} - 10 \, c^{4} d^{4} m^{3} + 35 \, c^{4} d^{4} m^{2} - 50 \, c^{4} d^{4} m + 24 \, c^{4} d^{4}\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 2024, normalized size = 5.90
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 527, normalized size = 1.54 \begin {gather*} -\frac {\left (c^{3} d^{3} g^{3} m^{3} x^{3}+3 c^{3} d^{3} f \,g^{2} m^{3} x^{2}-6 c^{3} d^{3} g^{3} m^{2} x^{3}+3 a \,c^{2} d^{2} e \,g^{3} m^{2} x^{2}+3 c^{3} d^{3} f^{2} g \,m^{3} x -21 c^{3} d^{3} f \,g^{2} m^{2} x^{2}+11 c^{3} d^{3} g^{3} m \,x^{3}+6 a \,c^{2} d^{2} e f \,g^{2} m^{2} x -9 a \,c^{2} d^{2} e \,g^{3} m \,x^{2}+c^{3} d^{3} f^{3} m^{3}-24 c^{3} d^{3} f^{2} g \,m^{2} x +42 c^{3} d^{3} f \,g^{2} m \,x^{2}-6 g^{3} x^{3} c^{3} d^{3}+6 a^{2} c d \,e^{2} g^{3} m x +3 a \,c^{2} d^{2} e \,f^{2} g \,m^{2}-30 a \,c^{2} d^{2} e f \,g^{2} m x +6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f^{3} m^{2}+57 c^{3} d^{3} f^{2} g m x -24 c^{3} d^{3} f \,g^{2} x^{2}+6 a^{2} c d \,e^{2} f \,g^{2} m -6 a^{2} c d \,e^{2} g^{3} x -21 a \,c^{2} d^{2} e \,f^{2} g m +24 a \,c^{2} d^{2} e f \,g^{2} x +26 c^{3} d^{3} f^{3} m -36 c^{3} d^{3} f^{2} g x +6 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+36 a \,c^{2} d^{2} e \,f^{2} g -24 f^{3} c^{3} d^{3}\right ) \left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{\left (m^{4}-10 m^{3}+35 m^{2}-50 m +24\right ) c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 331, normalized size = 0.97 \begin {gather*} -\frac {{\left (c d x + a e\right )} f^{3}}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} - \frac {3 \, {\left (c^{2} d^{2} {\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} f^{2} g}{{\left (m^{2} - 3 \, m + 2\right )} {\left (c d x + a e\right )}^{m} c^{2} d^{2}} - \frac {3 \, {\left ({\left (m^{2} - 3 \, m + 2\right )} c^{3} d^{3} x^{3} + {\left (m^{2} - m\right )} a c^{2} d^{2} e x^{2} + 2 \, a^{2} c d e^{2} m x + 2 \, a^{3} e^{3}\right )} f g^{2}}{{\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} {\left (c d x + a e\right )}^{m} c^{3} d^{3}} - \frac {{\left ({\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} c^{4} d^{4} x^{4} + {\left (m^{3} - 3 \, m^{2} + 2 \, m\right )} a c^{3} d^{3} e x^{3} + 3 \, {\left (m^{2} - m\right )} a^{2} c^{2} d^{2} e^{2} x^{2} + 6 \, a^{3} c d e^{3} m x + 6 \, a^{4} e^{4}\right )} g^{3}}{{\left (m^{4} - 10 \, m^{3} + 35 \, m^{2} - 50 \, m + 24\right )} {\left (c d x + a e\right )}^{m} c^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.75, size = 615, normalized size = 1.79 \begin {gather*} -\frac {\frac {g^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3-6\,m^2+11\,m-6\right )}{m^4-10\,m^3+35\,m^2-50\,m+24}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (6\,a^3\,c\,d\,e^3\,g^3\,m+6\,a^2\,c^2\,d^2\,e^2\,f\,g^2\,m^2-24\,a^2\,c^2\,d^2\,e^2\,f\,g^2\,m+3\,a\,c^3\,d^3\,e\,f^2\,g\,m^3-21\,a\,c^3\,d^3\,e\,f^2\,g\,m^2+36\,a\,c^3\,d^3\,e\,f^2\,g\,m+c^4\,d^4\,f^3\,m^3-9\,c^4\,d^4\,f^3\,m^2+26\,c^4\,d^4\,f^3\,m-24\,c^4\,d^4\,f^3\right )}{c^4\,d^4\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {a\,e\,{\left (d+e\,x\right )}^m\,\left (6\,a^3\,e^3\,g^3+6\,a^2\,c\,d\,e^2\,f\,g^2\,m-24\,a^2\,c\,d\,e^2\,f\,g^2+3\,a\,c^2\,d^2\,e\,f^2\,g\,m^2-21\,a\,c^2\,d^2\,e\,f^2\,g\,m+36\,a\,c^2\,d^2\,e\,f^2\,g+c^3\,d^3\,f^3\,m^3-9\,c^3\,d^3\,f^3\,m^2+26\,c^3\,d^3\,f^3\,m-24\,c^3\,d^3\,f^3\right )}{c^4\,d^4\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {3\,g\,x^2\,\left (m-1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^2\,g^2\,m+a\,c\,d\,e\,f\,g\,m^2-4\,a\,c\,d\,e\,f\,g\,m+c^2\,d^2\,f^2\,m^2-7\,c^2\,d^2\,f^2\,m+12\,c^2\,d^2\,f^2\right )}{c^2\,d^2\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {g^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (a\,e\,g\,m-12\,c\,d\,f+3\,c\,d\,f\,m\right )\,\left (m^2-3\,m+2\right )}{c\,d\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \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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