Optimal. Leaf size=92 \[ -\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d^3}{4 b^4 (a+b x)^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} -\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d^3}{4 b^4 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b x)^8} \, dx &=\int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^8}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^7}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^6}+\frac {d^3}{b^3 (a+b x)^5}\right ) \, dx\\ &=-\frac {(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac {d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac {3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac {d^3}{4 b^4 (a+b x)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 97, normalized size = 1.05 \begin {gather*} -\frac {a^3 d^3+a^2 b d^2 (4 c+7 d x)+a b^2 d \left (10 c^2+28 c d x+21 d^2 x^2\right )+b^3 \left (20 c^3+70 c^2 d x+84 c d^2 x^2+35 d^3 x^3\right )}{140 b^4 (a+b x)^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^3}{(a+b x)^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.46, size = 182, normalized size = 1.98 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \, {\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \, {\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.95, size = 114, normalized size = 1.24 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + 84 \, b^{3} c d^{2} x^{2} + 21 \, a b^{2} d^{3} x^{2} + 70 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 7 \, a^{2} b d^{3} x + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3}}{140 \, {\left (b x + a\right )}^{7} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 122, normalized size = 1.33 \begin {gather*} -\frac {d^{3}}{4 \left (b x +a \right )^{4} b^{4}}+\frac {3 \left (a d -b c \right ) d^{2}}{5 \left (b x +a \right )^{5} b^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}{2 \left (b x +a \right )^{6} b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{7 \left (b x +a \right )^{7} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.45, size = 182, normalized size = 1.98 \begin {gather*} -\frac {35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \, {\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \, {\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \, {\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 176, normalized size = 1.91 \begin {gather*} -\frac {\frac {a^3\,d^3+4\,a^2\,b\,c\,d^2+10\,a\,b^2\,c^2\,d+20\,b^3\,c^3}{140\,b^4}+\frac {d^3\,x^3}{4\,b}+\frac {d\,x\,\left (a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2\right )}{20\,b^3}+\frac {3\,d^2\,x^2\,\left (a\,d+4\,b\,c\right )}{20\,b^2}}{a^7+7\,a^6\,b\,x+21\,a^5\,b^2\,x^2+35\,a^4\,b^3\,x^3+35\,a^3\,b^4\,x^4+21\,a^2\,b^5\,x^5+7\,a\,b^6\,x^6+b^7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.12, size = 196, normalized size = 2.13 \begin {gather*} \frac {- a^{3} d^{3} - 4 a^{2} b c d^{2} - 10 a b^{2} c^{2} d - 20 b^{3} c^{3} - 35 b^{3} d^{3} x^{3} + x^{2} \left (- 21 a b^{2} d^{3} - 84 b^{3} c d^{2}\right ) + x \left (- 7 a^{2} b d^{3} - 28 a b^{2} c d^{2} - 70 b^{3} c^{2} d\right )}{140 a^{7} b^{4} + 980 a^{6} b^{5} x + 2940 a^{5} b^{6} x^{2} + 4900 a^{4} b^{7} x^{3} + 4900 a^{3} b^{8} x^{4} + 2940 a^{2} b^{9} x^{5} + 980 a b^{10} x^{6} + 140 b^{11} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________