Optimal. Leaf size=101 \[ -\frac {4 b^3 (2 n+1) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{3 a^2 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {96, 131} \[ -\frac {4 b^3 (2 n+1) (a+b x)^{n-1} (a-b x)^{1-n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)}-\frac {(a+b x)^{n+2} (a-b x)^{1-n}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 96
Rule 131
Rubi steps
\begin {align*} \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^4} \, dx &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}+\frac {(b (1+2 n)) \int \frac {(a-b x)^{-n} (a+b x)^{1+n}}{x^3} \, dx}{3 a}\\ &=-\frac {(a-b x)^{1-n} (a+b x)^{2+n}}{3 a^2 x^3}-\frac {4 b^3 (1+2 n) (a-b x)^{1-n} (a+b x)^{-1+n} \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )}{3 a^2 (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 88, normalized size = 0.87 \[ \frac {(a-b x)^{1-n} (a+b x)^{n-1} \left (4 b^3 (2 n+1) x^3 \, _2F_1\left (3,1-n;2-n;\frac {a-b x}{a+b x}\right )-(n-1) (a+b x)^3\right )}{3 a^2 (n-1) x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b x +a \right )^{-n} \left (b x +a \right )^{n +1}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n + 1}}{{\left (-b x + a\right )}^{n} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^{n+1}}{x^4\,{\left (a-b\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________