Optimal. Leaf size=64 \[ \frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45}
\begin {gather*} \frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^4} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x}\\ &=\frac {1}{2} a^2 c^2 x \sqrt {c x^2}+\frac {2}{3} a b c^2 x^2 \sqrt {c x^2}+\frac {1}{4} b^2 c^2 x^3 \sqrt {c x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 36, normalized size = 0.56 \begin {gather*} \frac {1}{12} c^2 x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 2.03, size = 31, normalized size = 0.48 \begin {gather*} \frac {\left (6 a^2+8 a b x+3 b^2 x^2\right ) {\left (c x^2\right )}^{\frac {5}{2}}}{12 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 32, normalized size = 0.50
method | result | size |
gosper | \(\frac {\left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{3}}\) | \(32\) |
default | \(\frac {\left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{12 x^{3}}\) | \(32\) |
risch | \(\frac {a^{2} c^{2} x \sqrt {c \,x^{2}}}{2}+\frac {2 a b \,c^{2} x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} c^{2} x^{3} \sqrt {c \,x^{2}}}{4}\) | \(53\) |
trager | \(\frac {c^{2} \left (3 b^{2} x^{3}+8 a b \,x^{2}+3 x^{2} b^{2}+6 a^{2} x +8 a b x +3 b^{2} x +6 a^{2}+8 a b +3 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.29, size = 40, normalized size = 0.62 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{2} c^{2} x^{3} + 8 \, a b c^{2} x^{2} + 6 \, a^{2} c^{2} x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.43, size = 49, normalized size = 0.77 \begin {gather*} \frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{2 x^{3}} + \frac {2 a b \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 49, normalized size = 0.77 \begin {gather*} \sqrt {c} \left (\frac {1}{2} a^{2} c^{2} x^{2} \mathrm {sign}\left (x\right )+\frac {1}{4} b^{2} c^{2} x^{4} \mathrm {sign}\left (x\right )+\frac {2}{3} a b c^{2} x^{3} \mathrm {sign}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________