Integrand size = 21, antiderivative size = 91 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 c^2}{a \sqrt {a x+b x^2}}-\frac {2 \left (\frac {d^2}{b}+\frac {2 c (b c-a d)}{a^2}\right ) x}{\sqrt {a x+b x^2}}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{3/2}} \] Output:
-2*c^2/a/(b*x^2+a*x)^(1/2)-2*(d^2/b+2*c*(-a*d+b*c)/a^2)*x/(b*x^2+a*x)^(1/2 )+2*d^2*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(3/2)
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=-\frac {2 \left (\sqrt {b} \left (2 b^2 c^2 x+a^2 d^2 x+a b c (c-2 d x)\right )+a^2 d^2 \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{a^2 b^{3/2} \sqrt {x (a+b x)}} \] Input:
Integrate[(c + d*x)^2/(a*x + b*x^2)^(3/2),x]
Output:
(-2*(Sqrt[b]*(2*b^2*c^2*x + a^2*d^2*x + a*b*c*(c - 2*d*x)) + a^2*d^2*Sqrt[ x]*Sqrt[a + b*x]*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]]))/(a^2*b^(3/2)*Sq rt[x*(a + b*x)])
Time = 0.43 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1164, 25, 27, 1160, 1091, 219}
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.
\(\displaystyle \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1164 |
\(\displaystyle -\frac {2 \int -\frac {d (a c+(2 b c-a d) x)}{\sqrt {b x^2+a x}}dx}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \int \frac {d (a c+(2 b c-a d) x)}{\sqrt {b x^2+a x}}dx}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 d \int \frac {a c+(2 b c-a d) x}{\sqrt {b x^2+a x}}dx}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {2 d \left (\frac {a^2 d \int \frac {1}{\sqrt {b x^2+a x}}dx}{2 b}+\frac {\sqrt {a x+b x^2} (2 b c-a d)}{b}\right )}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {2 d \left (\frac {a^2 d \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{b}+\frac {\sqrt {a x+b x^2} (2 b c-a d)}{b}\right )}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 d \left (\frac {a^2 d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{3/2}}+\frac {\sqrt {a x+b x^2} (2 b c-a d)}{b}\right )}{a^2}-\frac {2 (c+d x) (x (2 b c-a d)+a c)}{a^2 \sqrt {a x+b x^2}}\) |
Input:
Int[(c + d*x)^2/(a*x + b*x^2)^(3/2),x]
Output:
(-2*(c + d*x)*(a*c + (2*b*c - a*d)*x))/(a^2*Sqrt[a*x + b*x^2]) + (2*d*(((2 *b*c - a*d)*Sqrt[a*x + b*x^2])/b + (a^2*d*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b *x^2]])/b^(3/2)))/a^2
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
Time = 0.82 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {-2 a c \left (-2 d x +c \right ) b^{\frac {3}{2}}-4 b^{\frac {5}{2}} c^{2} x +2 \left (\operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) \sqrt {x \left (b x +a \right )}-\sqrt {b}\, x \right ) d^{2} a^{2}}{b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a^{2}}\) | \(82\) |
risch | \(-\frac {2 c^{2} \left (b x +a \right )}{a^{2} \sqrt {x \left (b x +a \right )}}+\frac {\frac {d^{2} a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}-\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{b^{2} a \left (x +\frac {a}{b}\right )}}{a}\) | \(123\) |
default | \(-\frac {2 c^{2} \left (2 b x +a \right )}{a^{2} \sqrt {b \,x^{2}+a x}}+d^{2} \left (-\frac {x}{b \sqrt {b \,x^{2}+a x}}-\frac {a \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )}{2 b}+\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}\right )+2 c d \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )\) | \(169\) |
Input:
int((d*x+c)^2/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/b^(3/2)/(x*(b*x+a))^(1/2)*(-a*c*(-2*d*x+c)*b^(3/2)-2*b^(5/2)*c^2*x+(arct anh((x*(b*x+a))^(1/2)/x/b^(1/2))*(x*(b*x+a))^(1/2)-b^(1/2)*x)*d^2*a^2)/a^2
Time = 0.15 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.67 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (a b^{2} c^{2} + {\left (2 \, b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{a^{2} b^{3} x^{2} + a^{3} b^{2} x}, -\frac {2 \, {\left ({\left (a^{2} b d^{2} x^{2} + a^{3} d^{2} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (a b^{2} c^{2} + {\left (2 \, b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{a^{2} b^{3} x^{2} + a^{3} b^{2} x}\right ] \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
[((a^2*b*d^2*x^2 + a^3*d^2*x)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)* sqrt(b)) - 2*(a*b^2*c^2 + (2*b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x)*sqrt(b* x^2 + a*x))/(a^2*b^3*x^2 + a^3*b^2*x), -2*((a^2*b*d^2*x^2 + a^3*d^2*x)*sqr t(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (a*b^2*c^2 + (2*b^3*c ^2 - 2*a*b^2*c*d + a^2*b*d^2)*x)*sqrt(b*x^2 + a*x))/(a^2*b^3*x^2 + a^3*b^2 *x)]
\[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (x \left (a + b x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**2/(b*x**2+a*x)**(3/2),x)
Output:
Integral((c + d*x)**2/(x*(a + b*x))**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=-\frac {4 \, b c^{2} x}{\sqrt {b x^{2} + a x} a^{2}} + \frac {4 \, c d x}{\sqrt {b x^{2} + a x} a} - \frac {2 \, d^{2} x}{\sqrt {b x^{2} + a x} b} + \frac {d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{b^{\frac {3}{2}}} - \frac {2 \, c^{2}}{\sqrt {b x^{2} + a x} a} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
-4*b*c^2*x/(sqrt(b*x^2 + a*x)*a^2) + 4*c*d*x/(sqrt(b*x^2 + a*x)*a) - 2*d^2 *x/(sqrt(b*x^2 + a*x)*b) + d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b) )/b^(3/2) - 2*c^2/(sqrt(b*x^2 + a*x)*a)
Time = 0.14 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=-\frac {d^{2} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{b^{\frac {3}{2}}} - \frac {2 \, {\left (\frac {c^{2}}{a} + \frac {{\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{a^{2} b}\right )}}{\sqrt {b x^{2} + a x}} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
-d^2*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(3/2) - 2*( c^2/a + (2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(a^2*b))/sqrt(b*x^2 + a*x)
Time = 9.83 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=\frac {d^2\,\ln \left (\frac {\frac {a}{2}+b\,x}{\sqrt {b}}+\sqrt {b\,x^2+a\,x}\right )}{b^{3/2}}-\frac {c^2\,\left (2\,a+4\,b\,x\right )}{a^2\,\sqrt {b\,x^2+a\,x}}-\frac {2\,d^2\,x}{b\,\sqrt {b\,x^2+a\,x}}+\frac {4\,c\,d\,x}{a\,\sqrt {x\,\left (a+b\,x\right )}} \] Input:
int((c + d*x)^2/(a*x + b*x^2)^(3/2),x)
Output:
(d^2*log((a/2 + b*x)/b^(1/2) + (a*x + b*x^2)^(1/2)))/b^(3/2) - (c^2*(2*a + 4*b*x))/(a^2*(a*x + b*x^2)^(1/2)) - (2*d^2*x)/(b*(a*x + b*x^2)^(1/2)) + ( 4*c*d*x)/(a*(x*(a + b*x))^(1/2))
Time = 0.24 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.63 \[ \int \frac {(c+d x)^2}{\left (a x+b x^2\right )^{3/2}} \, dx=\frac {2 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} d^{2} x -2 \sqrt {b}\, \sqrt {b x +a}\, a^{2} d^{2} x +4 \sqrt {b}\, \sqrt {b x +a}\, a b c d x -4 \sqrt {b}\, \sqrt {b x +a}\, b^{2} c^{2} x -2 \sqrt {x}\, a^{2} b \,d^{2} x -2 \sqrt {x}\, a \,b^{2} c^{2}+4 \sqrt {x}\, a \,b^{2} c d x -4 \sqrt {x}\, b^{3} c^{2} x}{\sqrt {b x +a}\, a^{2} b^{2} x} \] Input:
int((d*x+c)^2/(b*x^2+a*x)^(3/2),x)
Output:
(2*(sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a **2*d**2*x - sqrt(b)*sqrt(a + b*x)*a**2*d**2*x + 2*sqrt(b)*sqrt(a + b*x)*a *b*c*d*x - 2*sqrt(b)*sqrt(a + b*x)*b**2*c**2*x - sqrt(x)*a**2*b*d**2*x - s qrt(x)*a*b**2*c**2 + 2*sqrt(x)*a*b**2*c*d*x - 2*sqrt(x)*b**3*c**2*x))/(sqr t(a + b*x)*a**2*b**2*x)