Optimal. Leaf size=102 \[ -\frac {a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {772} \begin {gather*} -\frac {a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rubi steps
\begin {align*} \int x (a+b x)^n \left (c+d x^2\right ) \, dx &=\int \left (\frac {a \left (-b^2 c-a^2 d\right ) (a+b x)^n}{b^3}+\frac {\left (b^2 c+3 a^2 d\right ) (a+b x)^{1+n}}{b^3}-\frac {3 a d (a+b x)^{2+n}}{b^3}+\frac {d (a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=-\frac {a \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac {\left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a d (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d (a+b x)^{4+n}}{b^4 (4+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 109, normalized size = 1.07 \begin {gather*} \frac {(a+b x)^{n+1} \left (-6 a^3 d+6 a^2 b d (n+1) x-a b^2 \left (c \left (n^2+7 n+12\right )+3 d \left (n^2+3 n+2\right ) x^2\right )+b^3 \left (n^2+4 n+3\right ) x \left (c (n+4)+d (n+2) x^2\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int x (a+b x)^n \left (c+d x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 250, normalized size = 2.45 \begin {gather*} -\frac {{\left (a^{2} b^{2} c n^{2} + 7 \, a^{2} b^{2} c n + 12 \, a^{2} b^{2} c + 6 \, a^{4} d - {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} - {\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - {\left (b^{4} c n^{3} + 12 \, b^{4} c + {\left (8 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n^{2} + {\left (19 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{3} c n^{3} + 7 \, a b^{3} c n^{2} + 6 \, {\left (2 \, a b^{3} c + a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 410, normalized size = 4.02 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} x + 8 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} + 7 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} x + 19 \, {\left (b x + a\right )}^{n} b^{4} c n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} - {\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 12 \, {\left (b x + a\right )}^{n} b^{4} c x^{2} - 7 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c n - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 195, normalized size = 1.91 \begin {gather*} -\frac {\left (-b^{3} d \,n^{3} x^{3}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-b^{3} c \,n^{3} x -11 b^{3} d n \,x^{3}+9 a \,b^{2} d n \,x^{2}-8 b^{3} c \,n^{2} x -6 d \,x^{3} b^{3}-6 a^{2} b d n x +a \,b^{2} c \,n^{2}+6 a d \,x^{2} b^{2}-19 b^{3} c n x -6 a^{2} b d x +7 a \,b^{2} c n -12 b^{3} c x +6 a^{3} d +12 a \,b^{2} c \right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.47, size = 146, normalized size = 1.43 \begin {gather*} \frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.70, size = 255, normalized size = 2.50 \begin {gather*} {\left (a+b\,x\right )}^n\,\left (\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {a^2\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,d\,a^2\,n+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.53, size = 2181, normalized size = 21.38
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________