Optimal. Leaf size=63 \[ \frac {\sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac {\sqrt {c x^2} (a+b x)^{n+2}}{b^2 (n+2) x}-\frac {a \sqrt {c x^2} (a+b x)^{n+1}}{b^2 (n+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 43
Rubi steps
\begin {align*} \int \sqrt {c x^2} (a+b x)^n \, dx &=\frac {\sqrt {c x^2} \int x (a+b x)^n \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx}{x}\\ &=-\frac {a \sqrt {c x^2} (a+b x)^{1+n}}{b^2 (1+n) x}+\frac {\sqrt {c x^2} (a+b x)^{2+n}}{b^2 (2+n) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 44, normalized size = 0.70 \[ \frac {c x (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 63, normalized size = 1.00 \[ \frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.14, size = 119, normalized size = 1.89 \[ {\left (\frac {a^{2} a^{n} \mathrm {sgn}\relax (x)}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}} + \frac {{\left (b x + a\right )}^{n} b^{2} n x^{2} \mathrm {sgn}\relax (x) + {\left (b x + a\right )}^{n} a b n x \mathrm {sgn}\relax (x) + {\left (b x + a\right )}^{n} b^{2} x^{2} \mathrm {sgn}\relax (x) - {\left (b x + a\right )}^{n} a^{2} \mathrm {sgn}\relax (x)}{b^{2} n^{2} + 3 \, b^{2} n + 2 \, b^{2}}\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 46, normalized size = 0.73 \[ -\frac {\sqrt {c \,x^{2}}\, \left (-x n b -b x +a \right ) \left (b x +a \right )^{n +1}}{\left (n^{2}+3 n +2\right ) b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.45, size = 51, normalized size = 0.81 \[ \frac {{\left (b^{2} \sqrt {c} {\left (n + 1\right )} x^{2} + a b \sqrt {c} n x - a^{2} \sqrt {c}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.22, size = 85, normalized size = 1.35 \[ \frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{n^2+3\,n+2}-\frac {a^2\,\sqrt {c\,x^2}}{b^2\,\left (n^2+3\,n+2\right )}+\frac {a\,n\,x\,\sqrt {c\,x^2}}{b\,\left (n^2+3\,n+2\right )}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {a^{n} \sqrt {c} x \sqrt {x^{2}}}{2} & \text {for}\: b = 0 \\\int \frac {\sqrt {c x^{2}}}{\left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {\sqrt {c x^{2}}}{a + b x}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} \sqrt {c} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {a b \sqrt {c} n x \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {b^{2} \sqrt {c} n x^{2} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} + \frac {b^{2} \sqrt {c} x^{2} \left (a + b x\right )^{n} \sqrt {x^{2}}}{b^{2} n^{2} x + 3 b^{2} n x + 2 b^{2} x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________