Optimal. Leaf size=33 \[ \frac {c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, 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, 32} \[ \frac {c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^n}{x^5} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int (a+b x)^n \, dx}{x}\\ &=\frac {c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b (1+n) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 31, normalized size = 0.94 \[ \frac {c^3 x (a+b x)^{n+1}}{b (n+1) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 37, normalized size = 1.12 \[ \frac {{\left (b c^{2} x + a c^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x^{2}\right )^{\frac {5}{2}} {\left (b x + a\right )}^{n}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 29, normalized size = 0.88 \[ \frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x +a \right )^{n +1}}{\left (n +1\right ) b \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 28, normalized size = 0.85 \[ \frac {{\left (b c^{\frac {5}{2}} x + a c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{n}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.23, size = 49, normalized size = 1.48 \[ \frac {\left (\frac {c^2\,x\,\sqrt {c\,x^2}}{n+1}+\frac {a\,c^2\,\sqrt {c\,x^2}}{b\,\left (n+1\right )}\right )\,{\left (a+b\,x\right )}^n}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{a x^{4}} & \text {for}\: b = 0 \wedge n = -1 \\\frac {a^{n} c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}}{x^{4}} & \text {for}\: b = 0 \\\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{5} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\\frac {a c^{\frac {5}{2}} \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b n x^{5} + b x^{5}} + \frac {b c^{\frac {5}{2}} x \left (a + b x\right )^{n} \left (x^{2}\right )^{\frac {5}{2}}}{b n x^{5} + b x^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________