Optimal. Leaf size=56 \[ a^3 A x+a^2 A c x^3+\frac {3}{5} a A c^2 x^5+\frac {B \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} A c^3 x^7 \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 56, 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 = {641, 194} \begin {gather*} a^2 A c x^3+a^3 A x+\frac {3}{5} a A c^2 x^5+\frac {B \left (a+c x^2\right )^4}{8 c}+\frac {1}{7} A c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 194
Rule 641
Rubi steps
\begin {align*} \int (A+B x) \left (a+c x^2\right )^3 \, dx &=\frac {B \left (a+c x^2\right )^4}{8 c}+A \int \left (a+c x^2\right )^3 \, dx\\ &=\frac {B \left (a+c x^2\right )^4}{8 c}+A \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 A x+a^2 A c x^3+\frac {3}{5} a A c^2 x^5+\frac {1}{7} A c^3 x^7+\frac {B \left (a+c x^2\right )^4}{8 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 85, normalized size = 1.52 \begin {gather*} a^3 A x+\frac {1}{2} a^3 B x^2+a^2 A c x^3+\frac {3}{4} a^2 B c x^4+\frac {3}{5} a A c^2 x^5+\frac {1}{2} a B c^2 x^6+\frac {1}{7} A c^3 x^7+\frac {1}{8} B c^3 x^8 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) \left (a+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.36, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} x^{8} c^{3} B + \frac {1}{7} x^{7} c^{3} A + \frac {1}{2} x^{6} c^{2} a B + \frac {3}{5} x^{5} c^{2} a A + \frac {3}{4} x^{4} c a^{2} B + x^{3} c a^{2} A + \frac {1}{2} x^{2} a^{3} B + x a^{3} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 74, normalized size = 1.32 \begin {gather*} \frac {1}{8} B \,c^{3} x^{8}+\frac {1}{7} A \,c^{3} x^{7}+\frac {1}{2} B a \,c^{2} x^{6}+\frac {3}{5} A a \,c^{2} x^{5}+\frac {3}{4} B \,a^{2} c \,x^{4}+A \,a^{2} c \,x^{3}+\frac {1}{2} B \,a^{3} x^{2}+A \,a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 73, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, B c^{3} x^{8} + \frac {1}{7} \, A c^{3} x^{7} + \frac {1}{2} \, B a c^{2} x^{6} + \frac {3}{5} \, A a c^{2} x^{5} + \frac {3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 73, normalized size = 1.30 \begin {gather*} \frac {B\,a^3\,x^2}{2}+A\,a^3\,x+\frac {3\,B\,a^2\,c\,x^4}{4}+A\,a^2\,c\,x^3+\frac {B\,a\,c^2\,x^6}{2}+\frac {3\,A\,a\,c^2\,x^5}{5}+\frac {B\,c^3\,x^8}{8}+\frac {A\,c^3\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 85, normalized size = 1.52 \begin {gather*} A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________