Optimal. Leaf size=80 \[ \frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2041, 2027,
2039} \begin {gather*} \frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2027
Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int x^2 \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}-\frac {(4 b) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{9 c}\\ &=-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}+\frac {\left (8 b^2\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{63 c^2}\\ &=\frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 46, normalized size = 0.58 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (8 b^2-20 b c x^2+35 c^2 x^4\right )}{315 c^3 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 50, normalized size = 0.62
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (35 c^{2} x^{4}-20 b c \,x^{2}+8 b^{2}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{315 c^{3} x^{3}}\) | \(50\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (35 c^{2} x^{4}-20 b c \,x^{2}+8 b^{2}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{315 c^{3} x^{3}}\) | \(50\) |
trager | \(\frac {\left (35 c^{4} x^{8}+50 b \,c^{3} x^{6}+3 b^{2} c^{2} x^{4}-4 b^{3} c \,x^{2}+8 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{315 c^{3} x}\) | \(65\) |
risch | \(\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (35 c^{4} x^{8}+50 b \,c^{3} x^{6}+3 b^{2} c^{2} x^{4}-4 b^{3} c \,x^{2}+8 b^{4}\right )}{315 x \,c^{3}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 57, normalized size = 0.71 \begin {gather*} \frac {{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt {c x^{2} + b}}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 64, normalized size = 0.80 \begin {gather*} \frac {{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.62, size = 60, normalized size = 0.75 \begin {gather*} -\frac {8 \, b^{\frac {9}{2}} \mathrm {sgn}\left (x\right )}{315 \, c^{3}} + \frac {35 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 90 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} b \mathrm {sgn}\left (x\right ) + 63 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b^{2} \mathrm {sgn}\left (x\right )}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.19, size = 51, normalized size = 0.64 \begin {gather*} \frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}\,\left (8\,b^2-20\,b\,c\,x^2+35\,c^2\,x^4\right )}{315\,c^3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________