Optimal. Leaf size=50 \[ -\frac {b^2 \sqrt {c x^2} (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1) x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 65} \[ -\frac {b^2 \sqrt {c x^2} (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 65
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)^n}{x^4} \, dx &=\frac {\sqrt {c x^2} \int \frac {(a+b x)^n}{x^3} \, dx}{x}\\ &=-\frac {b^2 \sqrt {c x^2} (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 50, normalized size = 1.00 \[ -\frac {b^2 \sqrt {c x^2} (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \,x^{2}}\, \left (b x +a \right )^{n}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,x^2}\,{\left (a+b\,x\right )}^n}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2}} \left (a + b x\right )^{n}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________