Optimal. Leaf size=816 \[ -\frac{c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)+\frac{-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) f (m+1) n^2}-\frac{c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)-\frac{-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) f (m+1) n^2}+\frac{\left (c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) x^n+\left (b^2-2 a c\right ) \left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right )+a b c (b d-2 a e) (m-3 n+1)\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac{\left (c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right ) (f x)^{m+1}}{2 a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )^2} \]
[Out]
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Rubi [A] time = 9.34195, antiderivative size = 816, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)+\frac{-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) f (m+1) n^2}-\frac{c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)-\frac{-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) f (m+1) n^2}+\frac{\left (c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) x^n+\left (b^2-2 a c\right ) \left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right )+a b c (b d-2 a e) (m-3 n+1)\right ) (f x)^{m+1}}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac{\left (c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right ) (f x)^{m+1}}{2 a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )^2} \]
Antiderivative was successfully verified.
[In] Int[((f*x)^m*(d + e*x^n))/(a + b*x^n + c*x^(2*n))^3,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x)**m*(d+e*x**n)/(a+b*x**n+c*x**(2*n))**3,x)
[Out]
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Mathematica [B] time = 7.91735, size = 20515, normalized size = 25.14 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In] Integrate[((f*x)^m*(d + e*x^n))/(a + b*x^n + c*x^(2*n))^3,x]
[Out]
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Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) }{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x)^m*(d+e*x^n)/(a+b*x^n+c*x^(2*n))^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(f*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \,{\left (b c^{2} x^{n} + b^{2} c + a c^{2}\right )} x^{4 \, n} + 3 \,{\left (2 \, a b c x^{n} + a b^{2} + a^{2} c\right )} x^{2 \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(f*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x)**m*(d+e*x**n)/(a+b*x**n+c*x**(2*n))**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^n + d)*(f*x)^m/(c*x^(2*n) + b*x^n + a)^3,x, algorithm="giac")
[Out]