Optimal. Leaf size=23 \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 261} \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 372
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{\left (a+b (c+d x)^4\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (a+b x^4\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ -\frac {1}{8 b d \left (a+b (c+d x)^4\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.01, size = 172, normalized size = 7.48 \[ -\frac {1}{8 \, {\left (b^{3} d^{9} x^{8} + 8 \, b^{3} c d^{8} x^{7} + 28 \, b^{3} c^{2} d^{7} x^{6} + 56 \, b^{3} c^{3} d^{6} x^{5} + 2 \, {\left (35 \, b^{3} c^{4} + a b^{2}\right )} d^{5} x^{4} + 8 \, {\left (7 \, b^{3} c^{5} + a b^{2} c\right )} d^{4} x^{3} + 4 \, {\left (7 \, b^{3} c^{6} + 3 \, a b^{2} c^{2}\right )} d^{3} x^{2} + 8 \, {\left (b^{3} c^{7} + a b^{2} c^{3}\right )} d^{2} x + {\left (b^{3} c^{8} + 2 \, a b^{2} c^{4} + a^{2} b\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 21, normalized size = 0.91 \[ -\frac {1}{8 \, {\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 56, normalized size = 2.43 \[ -\frac {1}{8 \left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 21, normalized size = 0.91 \[ -\frac {1}{8 \, {\left ({\left (d x + c\right )}^{4} b + a\right )}^{2} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.35, size = 171, normalized size = 7.43 \[ -\frac {1}{8\,b\,d\,\left (x^4\,\left (70\,b^2\,c^4\,d^4+2\,a\,b\,d^4\right )+x^3\,\left (56\,b^2\,c^5\,d^3+8\,a\,b\,c\,d^3\right )+a^2+x\,\left (8\,d\,b^2\,c^7+8\,a\,d\,b\,c^3\right )+b^2\,c^8+x^2\,\left (28\,b^2\,c^6\,d^2+12\,a\,b\,c^2\,d^2\right )+b^2\,d^8\,x^8+2\,a\,b\,c^4+8\,b^2\,c\,d^7\,x^7+56\,b^2\,c^3\,d^5\,x^5+28\,b^2\,c^2\,d^6\,x^6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.42, size = 197, normalized size = 8.57 \[ - \frac {1}{8 a^{2} b d + 16 a b^{2} c^{4} d + 8 b^{3} c^{8} d + 448 b^{3} c^{3} d^{6} x^{5} + 224 b^{3} c^{2} d^{7} x^{6} + 64 b^{3} c d^{8} x^{7} + 8 b^{3} d^{9} x^{8} + x^{4} \left (16 a b^{2} d^{5} + 560 b^{3} c^{4} d^{5}\right ) + x^{3} \left (64 a b^{2} c d^{4} + 448 b^{3} c^{5} d^{4}\right ) + x^{2} \left (96 a b^{2} c^{2} d^{3} + 224 b^{3} c^{6} d^{3}\right ) + x \left (64 a b^{2} c^{3} d^{2} + 64 b^{3} c^{7} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________