Optimal. Leaf size=53 \[ 2 a A \sqrt {x}+\frac {2}{3} (A b+a B) x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779}
\begin {gather*} \frac {2}{3} x^{3/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{7} B c x^{7/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a A}{\sqrt {x}}+(A b+a B) \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=2 a A \sqrt {x}+\frac {2}{3} (A b+a B) x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 46, normalized size = 0.87 \begin {gather*} \frac {2}{105} \sqrt {x} (35 a (3 A+B x)+x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 40, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {2 \left (A b +B a \right ) x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}+2 a A \sqrt {x}\) | \(40\) |
default | \(\frac {2 \left (A b +B a \right ) x^{\frac {3}{2}}}{3}+\frac {2 \left (A c +B b \right ) x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}+2 a A \sqrt {x}\) | \(40\) |
trager | \(\left (\frac {2}{7} B c \,x^{3}+\frac {2}{5} A c \,x^{2}+\frac {2}{5} b B \,x^{2}+\frac {2}{3} A b x +\frac {2}{3} B a x +2 A a \right ) \sqrt {x}\) | \(41\) |
gosper | \(\frac {2 \sqrt {x}\, \left (15 B c \,x^{3}+21 A c \,x^{2}+21 b B \,x^{2}+35 A b x +35 B a x +105 A a \right )}{105}\) | \(42\) |
risch | \(\frac {2 \sqrt {x}\, \left (15 B c \,x^{3}+21 A c \,x^{2}+21 b B \,x^{2}+35 A b x +35 B a x +105 A a \right )}{105}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} + 2 \, A a \sqrt {x} + \frac {2}{3} \, {\left (B a + A b\right )} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.40, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 21 \, {\left (B b + A c\right )} x^{2} + 105 \, A a + 35 \, {\left (B a + A b\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.12, size = 68, normalized size = 1.28 \begin {gather*} 2 A a \sqrt {x} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.96, size = 43, normalized size = 0.81 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} + 2 \, A a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 41, normalized size = 0.77 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,c}{5}+\frac {2\,B\,b}{5}\right )+2\,A\,a\,\sqrt {x}+\frac {2\,B\,c\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________