Integrand size = 17, antiderivative size = 314 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=-\frac {1}{\left (a+b c^3\right ) x}+\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 d \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a+b c^3\right )^2}-\frac {3 b c^2 d \log (x)}{\left (a+b c^3\right )^2}+\frac {\sqrt [3]{b} \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) d \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}-\frac {\sqrt [3]{b} \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac {b c^2 d \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2} \]
-1/(b*c^3+a)/x-3*b*c^2*d*ln(x)/(b*c^3+a)^2+1/3*b^(1/3)*(a^(1/3)*(-2*b*c^3+ a)-b^(1/3)*c*(-b*c^3+2*a))*d*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(2/3)/(b*c^3+a) ^2-1/6*b^(1/3)*(a^(1/3)*(-2*b*c^3+a)-b^(1/3)*c*(-b*c^3+2*a))*d*ln(a^(2/3)- a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(2/3)/(b*c^3+a)^2+b*c^2*d*ln( a+b*(d*x+c)^3)/(b*c^3+a)^2+1/3*b^(1/3)*(a^(1/3)-b^(1/3)*c)*(a^(1/3)+b^(1/3 )*c)^3*d*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))/a^(1/3)*3^(1/2))/a^(2/3)/( b*c^3+a)^2*3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {-3 \left (a+b c^3+3 b c^2 d x \log (x)\right )+d x \text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {-3 a c \log (x-\text {$\#$1})+6 b c^4 \log (x-\text {$\#$1})-a d \log (x-\text {$\#$1}) \text {$\#$1}+8 b c^3 d \log (x-\text {$\#$1}) \text {$\#$1}+3 b c^2 d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 \left (a+b c^3\right )^2 x} \]
(-3*(a + b*c^3 + 3*b*c^2*d*x*Log[x]) + d*x*RootSum[a + b*c^3 + 3*b*c^2*d*# 1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & , (-3*a*c*Log[x - #1] + 6*b*c^4*Log[x - #1] - a*d*Log[x - #1]*#1 + 8*b*c^3*d*Log[x - #1]*#1 + 3*b*c^2*d^2*Log[x - #1]*#1^2)/(c^2 + 2*c*d*#1 + d^2*#1^2) & ])/(3*(a + b*c^3)^2*x)
Time = 0.70 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {896, 7276, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle d \int \frac {1}{d^2 x^2 \left (b (c+d x)^3+a\right )}d(c+d x)\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle d \int \left (-\frac {3 b c^2}{\left (b c^3+a\right )^2 d x}+\frac {b \left (3 b c^2 (c+d x)^2-\left (a-2 b c^3\right ) (c+d x)-c \left (2 a-b c^3\right )\right )}{\left (b c^3+a\right )^2 \left (b (c+d x)^3+a\right )}+\frac {1}{\left (b c^3+a\right ) d^2 x^2}\right )d(c+d x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle d \left (\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a+b c^3\right )^2}+\frac {b^{2/3} \left (-\frac {\sqrt [3]{a} \left (a-2 b c^3\right )}{\sqrt [3]{b}}+2 a c-b c^4\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac {\sqrt [3]{b} \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}-\frac {1}{d x \left (a+b c^3\right )}-\frac {3 b c^2 \log (-d x)}{\left (a+b c^3\right )^2}+\frac {b c^2 \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2}\right )\) |
d*(-(1/((a + b*c^3)*d*x)) + (b^(1/3)*(a^(1/3) - b^(1/3)*c)*(a^(1/3) + b^(1 /3)*c)^3*ArcTan[(a^(1/3) - 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/(Sqrt[ 3]*a^(2/3)*(a + b*c^3)^2) - (3*b*c^2*Log[-(d*x)])/(a + b*c^3)^2 + (b^(1/3) *(a^(1/3)*(a - 2*b*c^3) - b^(1/3)*c*(2*a - b*c^3))*Log[a^(1/3) + b^(1/3)*( c + d*x)])/(3*a^(2/3)*(a + b*c^3)^2) + (b^(2/3)*(2*a*c - b*c^4 - (a^(1/3)* (a - 2*b*c^3))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)* (c + d*x)^2])/(6*a^(2/3)*(a + b*c^3)^2) + (b*c^2*Log[a + b*(c + d*x)^3])/( a + b*c^3)^2)
3.2.8.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.46
method | result | size |
default | \(-\frac {1}{\left (b \,c^{3}+a \right ) x}-\frac {3 b \,c^{2} d \ln \left (x \right )}{\left (b \,c^{3}+a \right )^{2}}-\frac {d \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b \,c^{2} d^{2}-8 \textit {\_R} b \,c^{3} d -6 b \,c^{4}+\textit {\_R} a d +3 a c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{3 \left (b \,c^{3}+a \right )^{2}}\) | \(143\) |
risch | \(-\frac {1}{\left (b \,c^{3}+a \right ) x}-\frac {3 b \,c^{2} d \ln \left (x \right )}{c^{6} b^{2}+2 a b \,c^{3}+a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} b^{2} c^{6}+2 b \,c^{3} a^{3}+a^{4}\right ) \textit {\_Z}^{3}-9 a^{2} b \,c^{2} d \,\textit {\_Z}^{2}+6 a b c \,d^{2} \textit {\_Z} -b \,d^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2 a \,b^{3} c^{9} d -6 a^{3} b \,c^{3} d -4 a^{4} d \right ) \textit {\_R}^{3}+\left (-b^{3} c^{8} d^{2}+16 a \,b^{2} c^{5} d^{2}+17 a^{2} b \,c^{2} d^{2}\right ) \textit {\_R}^{2}+\left (-9 b^{2} c^{4} d^{3}-18 b c \,d^{3} a \right ) \textit {\_R} +3 b \,d^{4}\right ) x +\left (a \,b^{3} c^{10}+3 a^{2} b^{2} c^{7}+3 a^{3} b \,c^{4}+c \,a^{4}\right ) \textit {\_R}^{3}+\left (-b^{3} c^{9} d +6 a \,b^{2} c^{6} d +6 a^{2} b \,c^{3} d -a^{3} d \right ) \textit {\_R}^{2}+\left (-9 b^{2} c^{5} d^{2}-9 b \,c^{2} d^{2} a \right ) \textit {\_R} \right )\right )}{3}\) | \(315\) |
-1/(b*c^3+a)/x-3*b*c^2*d*ln(x)/(b*c^3+a)^2-1/3*d*sum((-3*_R^2*b*c^2*d^2-8* _R*b*c^3*d-6*b*c^4+_R*a*d+3*a*c)/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootO f(_Z^3*b*d^3+3*_Z^2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))/(b*c^3+a)^2
Result contains complex when optimal does not.
Time = 1.33 (sec) , antiderivative size = 8919, normalized size of antiderivative = 28.40 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} x^{2}} \,d x } \]
-3*b*c^2*d*log(x)/(b^2*c^6 + 2*a*b*c^3 + a^2) + b*d^2*integrate((3*b*c^2*d ^2*x^2 + 6*b*c^4 + (8*b*c^3 - a)*d*x - 3*a*c)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/(b^2*c^6 + 2*a*b*c^3 + a^2) - 1/((b*c^3 + a) *x)
\[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} x^{2}} \,d x } \]
Time = 9.90 (sec) , antiderivative size = 1588, normalized size of antiderivative = 5.06 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^3\right )} \, dx=\text {Too large to display} \]
symsum(log((b^4*d^12*x - 3*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27 *a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^2*a^3*b^3*d^ 9 - 3*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c ^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^2*b^6*c^9*d^9 - 9*root(27*a^2*b^2 *c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d ^2*z - b*d^3, z, k)*b^5*c^5*d^10 + 18*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c ^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^2 *a^2*b^4*c^3*d^9 + 27*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4* z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^3*a^3*b^4*c^4*d^8 + 27*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c ^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^3*a^2*b^5*c^7*d^8 - 9*root(27*a^2 *b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b *c*d^2*z - b*d^3, z, k)*a*b^4*c^2*d^10 - 9*root(27*a^2*b^2*c^6*z^3 + 54*a^ 3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)*b^5*c^4*d^11*x + 9*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4 *z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^3*a^4*b^3*c*d^8 + 18*root(27*a^2*b^2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^ 2*d*z^2 + 18*a*b*c*d^2*z - b*d^3, z, k)^2*a*b^5*c^6*d^9 + 9*root(27*a^2*b^ 2*c^6*z^3 + 54*a^3*b*c^3*z^3 + 27*a^4*z^3 - 81*a^2*b*c^2*d*z^2 + 18*a*b*c* d^2*z - b*d^3, z, k)^3*a*b^6*c^10*d^8 - 36*root(27*a^2*b^2*c^6*z^3 + 54...