Optimal. Leaf size=299 \[ \frac{b^3 x (a d (1-4 n)-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^4}-\frac{d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{2 a c^2 n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{d^2 x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^4}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}+\frac{d x (a d+2 b c)}{2 a c n (b c-a d)^2 \left (c+d x^n\right )^2} \]
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Rubi [A] time = 1.30478, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{b^3 x (a d (1-4 n)-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n (b c-a d)^4}-\frac{d x \left (-a^2 d^2 (1-2 n)+a b c d (1-6 n)-2 b^2 c^2 n\right )}{2 a c^2 n^2 (b c-a d)^3 \left (c+d x^n\right )}+\frac{d^2 x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^4}+\frac{b x}{a n (b c-a d) \left (a+b x^n\right ) \left (c+d x^n\right )^2}+\frac{d x (a d+2 b c)}{2 a c n (b c-a d)^2 \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^n)^2*(c + d*x^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(1/(a+b*x**n)**2/(c+d*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.535841, size = 233, normalized size = 0.78 \[ \frac{x \left (\frac{2 b^3 n (a d (1-4 n)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}+\frac{d^2 \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (4 n^2-5 n+1\right )+b^2 c^2 \left (12 n^2-7 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^3}+\frac{2 b^3 n (b c-a d)}{a \left (a+b x^n\right )}+\frac{d^2 (a d-b c) (a d (2 n-1)+b (c-6 c n))}{c^2 \left (c+d x^n\right )}+\frac{d^2 n (b c-a d)^2}{c \left (c+d x^n\right )^2}\right )}{2 n^2 (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^n)^2*(c + d*x^n)^3),x]
[Out]
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Maple [F] time = 0.187, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^2/(c+d*x^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(1/((b*x^n + a)^2*(d*x^n + c)^3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{2} d^{3} x^{5 \, n} + a^{2} c^{3} +{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{4 \, n} +{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3 \, n} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2 \, n} +{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)^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(1/(a+b*x**n)**2/(c+d*x**n)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^2*(d*x^n + c)^3),x, algorithm="giac")
[Out]