Integrand size = 21, antiderivative size = 108 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {d (8 b c-3 a d) \sqrt {a x+b x^2}}{4 b^2}+\frac {d^2 x \sqrt {a x+b x^2}}{2 b}+\frac {\left (8 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{5/2}} \] Output:
1/4*d*(-3*a*d+8*b*c)*(b*x^2+a*x)^(1/2)/b^2+1/2*d^2*x*(b*x^2+a*x)^(1/2)/b+1 /4*(3*a^2*d^2-8*a*b*c*d+8*b^2*c^2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^ (5/2)
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {\sqrt {b} d x (a+b x) (8 b c-3 a d+2 b d x)+\left (-8 b^2 c^2+8 a b c d-3 a^2 d^2\right ) \sqrt {x} \sqrt {a+b x} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{4 b^{5/2} \sqrt {x (a+b x)}} \] Input:
Integrate[(c + d*x)^2/Sqrt[a*x + b*x^2],x]
Output:
(Sqrt[b]*d*x*(a + b*x)*(8*b*c - 3*a*d + 2*b*d*x) + (-8*b^2*c^2 + 8*a*b*c*d - 3*a^2*d^2)*Sqrt[x]*Sqrt[a + b*x]*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x] ])/(4*b^(5/2)*Sqrt[x*(a + b*x)])
Time = 0.45 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1166, 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.
\(\displaystyle \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx\) |
\(\Big \downarrow \) 1166 |
\(\displaystyle \frac {\int \frac {c (4 b c-a d)+3 d (2 b c-a d) x}{2 \sqrt {b x^2+a x}}dx}{2 b}+\frac {d \sqrt {a x+b x^2} (c+d x)}{2 b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {c (4 b c-a d)+3 d (2 b c-a d) x}{\sqrt {b x^2+a x}}dx}{4 b}+\frac {d \sqrt {a x+b x^2} (c+d x)}{2 b}\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a x}}dx}{2 b}+\frac {3 d \sqrt {a x+b x^2} (2 b c-a d)}{b}}{4 b}+\frac {d \sqrt {a x+b x^2} (c+d x)}{2 b}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {\frac {\left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{b}+\frac {3 d \sqrt {a x+b x^2} (2 b c-a d)}{b}}{4 b}+\frac {d \sqrt {a x+b x^2} (c+d x)}{2 b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (3 a^2 d^2-8 a b c d+8 b^2 c^2\right )}{b^{3/2}}+\frac {3 d \sqrt {a x+b x^2} (2 b c-a d)}{b}}{4 b}+\frac {d \sqrt {a x+b x^2} (c+d x)}{2 b}\) |
Input:
Int[(c + d*x)^2/Sqrt[a*x + b*x^2],x]
Output:
(d*(c + d*x)*Sqrt[a*x + b*x^2])/(2*b) + ((3*d*(2*b*c - a*d)*Sqrt[a*x + b*x ^2])/b + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/b^(3/2))/(4*b)
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[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[1/(c*(m + 2*p + 1)) Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* (a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
Time = 0.49 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {8}{3} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{4}-\frac {3 d \left (\frac {2 \left (-d x -4 c \right ) b^{\frac {3}{2}}}{3}+\sqrt {b}\, a d \right ) \sqrt {x \left (b x +a \right )}}{4}}{b^{\frac {5}{2}}}\) | \(79\) |
risch | \(-\frac {\left (-2 b d x +3 a d -8 b c \right ) d x \left (b x +a \right )}{4 b^{2} \sqrt {x \left (b x +a \right )}}+\frac {\left (3 a^{2} d^{2}-8 a b c d +8 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {5}{2}}}\) | \(89\) |
default | \(\frac {c^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}+d^{2} \left (\frac {x \sqrt {b \,x^{2}+a x}}{2 b}-\frac {3 a \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+2 c d \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )\) | \(157\) |
Input:
int((d*x+c)^2/(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
3/4/b^(5/2)*((a^2*d^2-8/3*a*b*c*d+8/3*b^2*c^2)*arctanh((x*(b*x+a))^(1/2)/x /b^(1/2))-d*(2/3*(-d*x-4*c)*b^(3/2)+b^(1/2)*a*d)*(x*(b*x+a))^(1/2))
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.77 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\left [\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x + 8 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{8 \, b^{3}}, -\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (2 \, b^{2} d^{2} x + 8 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{4 \, b^{3}}\right ] \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(1/2),x, algorithm="fricas")
Output:
[1/8*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*sqrt(b)*log(2*b*x + a + 2*sqrt(b *x^2 + a*x)*sqrt(b)) + 2*(2*b^2*d^2*x + 8*b^2*c*d - 3*a*b*d^2)*sqrt(b*x^2 + a*x))/b^3, -1/4*((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*sqrt(-b)*arctan(sqr t(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (2*b^2*d^2*x + 8*b^2*c*d - 3*a*b*d^2) *sqrt(b*x^2 + a*x))/b^3]
Time = 0.51 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.71 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\begin {cases} \sqrt {a x + b x^{2}} \left (\frac {d^{2} x}{2 b} + \frac {- \frac {3 a d^{2}}{4 b} + 2 c d}{b}\right ) + \left (- \frac {a \left (- \frac {3 a d^{2}}{4 b} + 2 c d\right )}{2 b} + c^{2}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (c^{2} \sqrt {a x} + \frac {2 c d \left (a x\right )^{\frac {3}{2}}}{3 a} + \frac {d^{2} \left (a x\right )^{\frac {5}{2}}}{5 a^{2}}\right )}{a} & \text {for}\: a \neq 0 \\\tilde {\infty } \left (\begin {cases} c^{2} x & \text {for}\: d = 0 \\\frac {\left (c + d x\right )^{3}}{3 d} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2/(b*x**2+a*x)**(1/2),x)
Output:
Piecewise((sqrt(a*x + b*x**2)*(d**2*x/(2*b) + (-3*a*d**2/(4*b) + 2*c*d)/b) + (-a*(-3*a*d**2/(4*b) + 2*c*d)/(2*b) + c**2)*Piecewise((log(a + 2*sqrt(b )*sqrt(a*x + b*x**2) + 2*b*x)/sqrt(b), Ne(a**2/b, 0)), ((a/(2*b) + x)*log( a/(2*b) + x)/sqrt(b*(a/(2*b) + x)**2), True)), Ne(b, 0)), (2*(c**2*sqrt(a* x) + 2*c*d*(a*x)**(3/2)/(3*a) + d**2*(a*x)**(5/2)/(5*a**2))/a, Ne(a, 0)), (zoo*Piecewise((c**2*x, Eq(d, 0)), ((c + d*x)**3/(3*d), True)), True))
Time = 0.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.43 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a x} d^{2} x}{2 \, b} + \frac {c^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{\sqrt {b}} - \frac {a c d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{b^{\frac {3}{2}}} + \frac {3 \, a^{2} d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {5}{2}}} + \frac {2 \, \sqrt {b x^{2} + a x} c d}{b} - \frac {3 \, \sqrt {b x^{2} + a x} a d^{2}}{4 \, b^{2}} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(1/2),x, algorithm="maxima")
Output:
1/2*sqrt(b*x^2 + a*x)*d^2*x/b + c^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sq rt(b))/sqrt(b) - a*c*d*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(3/2 ) + 3/8*a^2*d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(5/2) + 2*s qrt(b*x^2 + a*x)*c*d/b - 3/4*sqrt(b*x^2 + a*x)*a*d^2/b^2
Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (\frac {2 \, d^{2} x}{b} + \frac {8 \, b c d - 3 \, a d^{2}}{b^{2}}\right )} - \frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{8 \, b^{\frac {5}{2}}} \] Input:
integrate((d*x+c)^2/(b*x^2+a*x)^(1/2),x, algorithm="giac")
Output:
1/4*sqrt(b*x^2 + a*x)*(2*d^2*x/b + (8*b*c*d - 3*a*d^2)/b^2) - 1/8*(8*b^2*c ^2 - 8*a*b*c*d + 3*a^2*d^2)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt (b) + a))/b^(5/2)
Timed out. \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{\sqrt {b\,x^2+a\,x}} \,d x \] Input:
int((c + d*x)^2/(a*x + b*x^2)^(1/2),x)
Output:
int((c + d*x)^2/(a*x + b*x^2)^(1/2), x)
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x)^2}{\sqrt {a x+b x^2}} \, dx=\frac {-3 \sqrt {x}\, \sqrt {b x +a}\, a b \,d^{2}+8 \sqrt {x}\, \sqrt {b x +a}\, b^{2} c d +2 \sqrt {x}\, \sqrt {b x +a}\, b^{2} d^{2} x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} d^{2}-8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a b c d +8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{2} c^{2}}{4 b^{3}} \] Input:
int((d*x+c)^2/(b*x^2+a*x)^(1/2),x)
Output:
( - 3*sqrt(x)*sqrt(a + b*x)*a*b*d**2 + 8*sqrt(x)*sqrt(a + b*x)*b**2*c*d + 2*sqrt(x)*sqrt(a + b*x)*b**2*d**2*x + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt( x)*sqrt(b))/sqrt(a))*a**2*d**2 - 8*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sq rt(b))/sqrt(a))*a*b*c*d + 8*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/ sqrt(a))*b**2*c**2)/(4*b**3)