Integrand size = 20, antiderivative size = 58 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=-\frac {a \sqrt {c+d x}}{x}+\frac {2 b (c+d x)^{3/2}}{3 d}-\frac {a d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}} \] Output:
-a*(d*x+c)^(1/2)/x+2/3*b*(d*x+c)^(3/2)/d-a*d*arctanh((d*x+c)^(1/2)/c^(1/2) )/c^(1/2)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=-\frac {a \sqrt {c+d x}}{x}+\frac {2 b (c+d x)^{3/2}}{3 d}-\frac {a d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}} \] Input:
Integrate[(Sqrt[c + d*x]*(a + b*x^2))/x^2,x]
Output:
-((a*Sqrt[c + d*x])/x) + (2*b*(c + d*x)^(3/2))/(3*d) - (a*d*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c]
Time = 0.41 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {517, 1580, 25, 1467, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right ) \sqrt {c+d x}}{x^2} \, dx\) |
\(\Big \downarrow \) 517 |
\(\displaystyle \frac {2 \int \frac {(c+d x) \left (b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2\right )}{d^2 x^2}d\sqrt {c+d x}}{d}\) |
\(\Big \downarrow \) 1580 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {a d^2+2 b (c+d x)^2-2 b c (c+d x)}{d x}d\sqrt {c+d x}-\frac {a d \sqrt {c+d x}}{2 x}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \left (-\frac {1}{2} \int -\frac {a d^2+2 b (c+d x)^2-2 b c (c+d x)}{d x}d\sqrt {c+d x}-\frac {a d \sqrt {c+d x}}{2 x}\right )}{d}\) |
\(\Big \downarrow \) 1467 |
\(\displaystyle \frac {2 \left (-\frac {1}{2} \int \left (-\frac {a d}{x}-2 b (c+d x)\right )d\sqrt {c+d x}-\frac {a d \sqrt {c+d x}}{2 x}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {2}{3} b (c+d x)^{3/2}-\frac {a d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {a d \sqrt {c+d x}}{2 x}\right )}{d}\) |
Input:
Int[(Sqrt[c + d*x]*(a + b*x^2))/x^2,x]
Output:
(2*(-1/2*(a*d*Sqrt[c + d*x])/x + ((2*b*(c + d*x)^(3/2))/3 - (a*d^2*ArcTanh [Sqrt[c + d*x]/Sqrt[c]])/Sqrt[c])/2))/d
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*(e^m/d^(m + 2*p + 1)) Subst[Int[x^(2*n + 1)*(-c + x^ 2)^m*(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4)^p, x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[m, 0] && IntegerQ[n + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_) ^4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[1/(2*e^(2*p + m/2)* (q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2* e^(2*p + m/2)*(q + 1)*x^m*(a + b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b *d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b, c, d, e }, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {a \sqrt {d x +c}}{x}+\frac {\frac {2 b \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {a \,d^{2} \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}}{d}\) | \(51\) |
derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} \left (-\frac {\sqrt {d x +c}}{2 d x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{d}\) | \(55\) |
default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} \left (-\frac {\sqrt {d x +c}}{2 d x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{d}\) | \(55\) |
pseudoelliptic | \(-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right ) a \,d^{2} x +\left (-\frac {2 c^{\frac {3}{2}} b x}{3}+d \sqrt {c}\, \left (-\frac {2 b \,x^{2}}{3}+a \right )\right ) \sqrt {d x +c}}{\sqrt {c}\, d x}\) | \(60\) |
Input:
int((d*x+c)^(1/2)*(b*x^2+a)/x^2,x,method=_RETURNVERBOSE)
Output:
-a*(d*x+c)^(1/2)/x+1/d*(2/3*b*(d*x+c)^(3/2)-a*d^2/c^(1/2)*arctanh((d*x+c)^ (1/2)/c^(1/2)))
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=\left [\frac {3 \, a \sqrt {c} d^{2} x \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, b c d x^{2} + 2 \, b c^{2} x - 3 \, a c d\right )} \sqrt {d x + c}}{6 \, c d x}, \frac {3 \, a \sqrt {-c} d^{2} x \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + {\left (2 \, b c d x^{2} + 2 \, b c^{2} x - 3 \, a c d\right )} \sqrt {d x + c}}{3 \, c d x}\right ] \] Input:
integrate((d*x+c)^(1/2)*(b*x^2+a)/x^2,x, algorithm="fricas")
Output:
[1/6*(3*a*sqrt(c)*d^2*x*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*( 2*b*c*d*x^2 + 2*b*c^2*x - 3*a*c*d)*sqrt(d*x + c))/(c*d*x), 1/3*(3*a*sqrt(- c)*d^2*x*arctan(sqrt(-c)/sqrt(d*x + c)) + (2*b*c*d*x^2 + 2*b*c^2*x - 3*a*c *d)*sqrt(d*x + c))/(c*d*x)]
Time = 17.49 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=- \frac {a \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{\sqrt {x}} - \frac {a d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{\sqrt {c}} + b \left (\begin {cases} \frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} & \text {for}\: d \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) \] Input:
integrate((d*x+c)**(1/2)*(b*x**2+a)/x**2,x)
Output:
-a*sqrt(d)*sqrt(c/(d*x) + 1)/sqrt(x) - a*d*asinh(sqrt(c)/(sqrt(d)*sqrt(x)) )/sqrt(c) + b*Piecewise((2*(c + d*x)**(3/2)/(3*d), Ne(d, 0)), (sqrt(c)*x, True))
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=\frac {1}{6} \, {\left (\frac {3 \, a \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{\sqrt {c}} + \frac {4 \, {\left (d x + c\right )}^{\frac {3}{2}} b}{d^{2}} - \frac {6 \, \sqrt {d x + c} a}{d x}\right )} d \] Input:
integrate((d*x+c)^(1/2)*(b*x^2+a)/x^2,x, algorithm="maxima")
Output:
1/6*(3*a*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c)))/sqrt(c) + 4*(d*x + c)^(3/2)*b/d^2 - 6*sqrt(d*x + c)*a/(d*x))*d
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=\frac {1}{3} \, {\left (\frac {3 \, a \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b}{d^{2}} - \frac {3 \, \sqrt {d x + c} a}{d x}\right )} d \] Input:
integrate((d*x+c)^(1/2)*(b*x^2+a)/x^2,x, algorithm="giac")
Output:
1/3*(3*a*arctan(sqrt(d*x + c)/sqrt(-c))/sqrt(-c) + 2*(d*x + c)^(3/2)*b/d^2 - 3*sqrt(d*x + c)*a/(d*x))*d
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=\frac {2\,b\,{\left (c+d\,x\right )}^{3/2}}{3\,d}-\frac {a\,\sqrt {c+d\,x}}{x}-\frac {a\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )}{\sqrt {c}} \] Input:
int(((a + b*x^2)*(c + d*x)^(1/2))/x^2,x)
Output:
(2*b*(c + d*x)^(3/2))/(3*d) - (a*(c + d*x)^(1/2))/x - (a*d*atanh((c + d*x) ^(1/2)/c^(1/2)))/c^(1/2)
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )}{x^2} \, dx=\frac {-6 \sqrt {d x +c}\, a c d +4 \sqrt {d x +c}\, b \,c^{2} x +4 \sqrt {d x +c}\, b c d \,x^{2}+3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2} x -3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2} x}{6 c d x} \] Input:
int((d*x+c)^(1/2)*(b*x^2+a)/x^2,x)
Output:
( - 6*sqrt(c + d*x)*a*c*d + 4*sqrt(c + d*x)*b*c**2*x + 4*sqrt(c + d*x)*b*c *d*x**2 + 3*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*d**2*x - 3*sqrt(c)*log( sqrt(c + d*x) + sqrt(c))*a*d**2*x)/(6*c*d*x)