Optimal. Leaf size=80 \[ \frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {656, 648} \begin {gather*} \frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 648
Rule 656
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {b x+c x^2} \, dx &=\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {(4 b) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{7 c}\\ &=-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{35 c^2}\\ &=\frac {16 b^2 \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {8 b \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 42, normalized size = 0.52 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (8 b^2-12 b c x+15 c^2 x^2\right )}{105 c^3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.09, size = 55, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (8 b^3-4 b^2 c x+3 b c^2 x^2+15 c^3 x^3\right )}{105 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 49, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x^{2} + b x}}{105 \, c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 46, normalized size = 0.58 \begin {gather*} -\frac {16 \, b^{\frac {7}{2}}}{105 \, c^{3}} + \frac {2 \, {\left (15 \, {\left (c x + b\right )}^{\frac {7}{2}} - 42 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}\right )}}{105 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 44, normalized size = 0.55 \begin {gather*} \frac {2 \left (c x +b \right ) \left (15 c^{2} x^{2}-12 b c x +8 b^{2}\right ) \sqrt {c \,x^{2}+b x}}{105 c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.45, size = 42, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x + b}}{105 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{3/2}\,\sqrt {c\,x^2+b\,x} \,d 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^{\frac {3}{2}} \sqrt {x \left (b + c x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________