Optimal. Leaf size=38 \[ -\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac {\left (a+c x^4\right )^{5/2}}{10 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^7 \sqrt {a+c x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x \sqrt {a+c x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac {\left (a+c x^4\right )^{5/2}}{10 c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^4} \left (-2 a^2+a c x^4+3 c^2 x^8\right )}{30 c^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 25, normalized size = 0.66
method | result | size |
gosper | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\) | \(25\) |
default | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\) | \(25\) |
elliptic | \(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}} \left (-3 x^{4} c +2 a \right )}{30 c^{2}}\) | \(25\) |
trager | \(-\frac {\left (-3 c^{2} x^{8}-a c \,x^{4}+2 a^{2}\right ) \sqrt {x^{4} c +a}}{30 c^{2}}\) | \(36\) |
risch | \(-\frac {\left (-3 c^{2} x^{8}-a c \,x^{4}+2 a^{2}\right ) \sqrt {x^{4} c +a}}{30 c^{2}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{6 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (3 \, c^{2} x^{8} + a c x^{4} - 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{30 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.18, size = 61, normalized size = 1.61 \begin {gather*} \begin {cases} - \frac {a^{2} \sqrt {a + c x^{4}}}{15 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{4}}}{30 c} + \frac {x^{8} \sqrt {a + c x^{4}}}{10} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.57, size = 29, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} - 5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{30 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.13, size = 33, normalized size = 0.87 \begin {gather*} \sqrt {c\,x^4+a}\,\left (\frac {x^8}{10}-\frac {a^2}{15\,c^2}+\frac {a\,x^4}{30\,c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________