Integrand size = 28, antiderivative size = 204 \[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-d e+c f} \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{(b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}} \sqrt {e+f x}} \]
-2*b*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(1/2)+2*Ell ipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),(-b*(-c*f+d*e)/(-a*d+b*c)/f)^ (1/2))*f^(1/2)*(c*f-d*e)^(1/2)*(b*x+a)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)/ (-a*d+b*c)/(-a*f+b*e)/(-d*(b*x+a)/(-a*d+b*c))^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 15.72 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \left (-1-\frac {i \sqrt {\frac {d (a+b x)}{b (c+d x)}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right )|\frac {b c f-a d f}{b d e-a d f}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d (a+b x)}{b c-a d}}\right ),\frac {b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt {\frac {b (e+f x)}{b e-a f}}}\right )}{(b c-a d) (b e-a f) \sqrt {a+b x}} \]
(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-1 - (I*Sqrt[(d*(a + b*x))/(b*(c + d*x)) ]*(EllipticE[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]], (b*c*f - a*d*f)/( b*d*e - a*d*f)] - EllipticF[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]], (b *c*f - a*d*f)/(b*d*e - a*d*f)]))/Sqrt[(b*(e + f*x))/(b*e - a*f)]))/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x])
Time = 0.31 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {115, 27, 124, 123}
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.
| \(\displaystyle \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int -\frac {d f \sqrt {a+b x}}{2 \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d f \int \frac {\sqrt {a+b x}}{\sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {d f \sqrt {a+b x} \sqrt {\frac {d (e+f x)}{d e-c f}} \int \frac {\sqrt {-\frac {b x d}{b c-a d}-\frac {a d}{b c-a d}}}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}}}dx}{\sqrt {e+f x} (b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2 \sqrt {f} \sqrt {a+b x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|-\frac {b (d e-c f)}{(b c-a d) f}\right )}{\sqrt {e+f x} (b c-a d) (b e-a f) \sqrt {-\frac {d (a+b x)}{b c-a d}}}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d) (b e-a f)}\) |
(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c *f)]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*(d *e - c*f))/((b*c - a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x) )/(b*c - a*d))]*Sqrt[e + f*x])
3.29.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(564\) vs. \(2(181)=362\).
Time = 4.55 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.77
| method | result | size |
| default | \(-\frac {2 \left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, E\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a c \,f^{2}-\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, E\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) a d e f -\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, E\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b c e f +\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}\, \sqrt {\frac {\left (b x +a \right ) f}{a f -b e}}\, \sqrt {\frac {f \left (d x +c \right )}{c f -d e}}\, E\left (\sqrt {-\frac {d \left (f x +e \right )}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) b}{d \left (a f -b e \right )}}\right ) b d \,e^{2}+b d \,f^{2} x^{2}+b c \,f^{2} x +b d e f x +b c e f \right ) \sqrt {f x +e}\, \sqrt {d x +c}\, \sqrt {b x +a}}{f \left (a d -b c \right ) \left (a f -b e \right ) \left (b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e \right )}\) | \(565\) |
| elliptic | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (-\frac {2 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}{\left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (\frac {a d f -b c f -b d e}{a^{2} d f -a c f b -a b d e +b^{2} c e}+\frac {b c f +b d e}{a^{2} d f -a c f b -a b d e +b^{2} c e}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {2 b d f \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) E\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\) | \(690\) |
-2*((-d*(f*x+e)/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1/2)*(f*(d*x+c)/(c *f-d*e))^(1/2)*EllipticE((-d*(f*x+e)/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f- b*e))^(1/2))*a*c*f^2-(-d*(f*x+e)/(c*f-d*e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1 /2)*(f*(d*x+c)/(c*f-d*e))^(1/2)*EllipticE((-d*(f*x+e)/(c*f-d*e))^(1/2),((c *f-d*e)*b/d/(a*f-b*e))^(1/2))*a*d*e*f-(-d*(f*x+e)/(c*f-d*e))^(1/2)*((b*x+a )*f/(a*f-b*e))^(1/2)*(f*(d*x+c)/(c*f-d*e))^(1/2)*EllipticE((-d*(f*x+e)/(c* f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*c*e*f+(-d*(f*x+e)/(c*f-d* e))^(1/2)*((b*x+a)*f/(a*f-b*e))^(1/2)*(f*(d*x+c)/(c*f-d*e))^(1/2)*Elliptic E((-d*(f*x+e)/(c*f-d*e))^(1/2),((c*f-d*e)*b/d/(a*f-b*e))^(1/2))*b*d*e^2+b* d*f^2*x^2+b*c*f^2*x+b*d*e*f*x+b*c*e*f)*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a) ^(1/2)/f/(a*d-b*c)/(a*f-b*e)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c* f*x+a*d*e*x+b*c*e*x+a*c*e)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 812, normalized size of antiderivative = 3.98 \[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} b^{2} d f + {\left (a b d e + {\left (a b c - 2 \, a^{2} d\right )} f + {\left (b^{2} d e + {\left (b^{2} c - 2 \, a b d\right )} f\right )} x\right )} \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right ) + 3 \, {\left (b^{2} d f x + a b d f\right )} \sqrt {b d f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )\right )}}{3 \, {\left ({\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} e f - {\left (a^{2} b^{2} c d - a^{3} b d^{2}\right )} f^{2} + {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} e f - {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} f^{2}\right )} x\right )}} \]
-2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*b^2*d*f + (a*b*d*e + (a* b*c - 2*a^2*d)*f + (b^2*d*e + (b^2*c - 2*a*b*d)*f)*x)*sqrt(b*d*f)*weierstr assPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c* d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b ^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c ^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3 *b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(b^2*d*f*x + a*b*d*f)*sqrt( b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c ^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3* c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f ^3), weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2 *c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^ 3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f ^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3 *f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))))/((a*b^3*c*d - a^ 2*b^2*d^2)*e*f - (a^2*b^2*c*d - a^3*b*d^2)*f^2 + ((b^4*c*d - a*b^3*d^2)*e* f - (a*b^3*c*d - a^2*b^2*d^2)*f^2)*x)
\[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \]
\[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
\[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]