Integrand size = 20, antiderivative size = 80 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=-\frac {a \sqrt {c+d x}}{2 c x^2}+\frac {3 a d \sqrt {c+d x}}{4 c^2 x}-\frac {\left (8 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{5/2}} \] Output:
-1/2*a*(d*x+c)^(1/2)/c/x^2+3/4*a*d*(d*x+c)^(1/2)/c^2/x-1/4*(3*a*d^2+8*b*c^ 2)*arctanh((d*x+c)^(1/2)/c^(1/2))/c^(5/2)
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\frac {a \sqrt {c+d x} (-2 c+3 d x)}{4 c^2 x^2}-\frac {\left (8 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{5/2}} \] Input:
Integrate[(a + b*x^2)/(x^3*Sqrt[c + d*x]),x]
Output:
(a*Sqrt[c + d*x]*(-2*c + 3*d*x))/(4*c^2*x^2) - ((8*b*c^2 + 3*a*d^2)*ArcTan h[Sqrt[c + d*x]/Sqrt[c]])/(4*c^(5/2))
Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {517, 25, 1471, 25, 298, 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.
\(\displaystyle \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 517 |
\(\displaystyle 2 \int \frac {b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2}{d^3 x^3}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int -\frac {b c^2-2 b (c+d x) c+a d^2+b (c+d x)^2}{d^3 x^3}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 1471 |
\(\displaystyle 2 \left (\frac {\int -\frac {4 b c^2-4 b (c+d x) c+3 a d^2}{d^2 x^2}d\sqrt {c+d x}}{4 c}-\frac {a \sqrt {c+d x}}{4 c x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (-\frac {\int \frac {4 b c^2-4 b (c+d x) c+3 a d^2}{d^2 x^2}d\sqrt {c+d x}}{4 c}-\frac {a \sqrt {c+d x}}{4 c x^2}\right )\) |
\(\Big \downarrow \) 298 |
\(\displaystyle 2 \left (-\frac {\frac {\left (3 a d^2+8 b c^2\right ) \int -\frac {1}{d x}d\sqrt {c+d x}}{2 c}-\frac {3 a d \sqrt {c+d x}}{2 c x}}{4 c}-\frac {a \sqrt {c+d x}}{4 c x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (-\frac {\frac {\left (3 a d^2+8 b c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {3 a d \sqrt {c+d x}}{2 c x}}{4 c}-\frac {a \sqrt {c+d x}}{4 c x^2}\right )\) |
Input:
Int[(a + b*x^2)/(x^3*Sqrt[c + d*x]),x]
Output:
2*(-1/4*(a*Sqrt[c + d*x])/(c*x^2) - ((-3*a*d*Sqrt[c + d*x])/(2*c*x) + ((8* b*c^2 + 3*a*d^2)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(3/2)))/(4*c))
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
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] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], 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] && LtQ[q, -1]
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\sqrt {d x +c}\, a \left (-3 d x +2 c \right )}{4 c^{2} x^{2}}-\frac {\left (3 a \,d^{2}+8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4 c^{\frac {5}{2}}}\) | \(56\) |
pseudoelliptic | \(\frac {-\frac {3 x^{2} \left (a \,d^{2}+\frac {8 b \,c^{2}}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4}+\frac {3 a \sqrt {d x +c}\, \left (d x \sqrt {c}-\frac {2 c^{\frac {3}{2}}}{3}\right )}{4}}{c^{\frac {5}{2}} x^{2}}\) | \(60\) |
derivativedivides | \(-\frac {2 \left (-\frac {3 a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8 c^{2}}+\frac {5 a \,d^{2} \sqrt {d x +c}}{8 c}\right )}{d^{2} x^{2}}-\frac {\left (3 a \,d^{2}+8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4 c^{\frac {5}{2}}}\) | \(73\) |
default | \(-\frac {2 \left (-\frac {3 a \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{8 c^{2}}+\frac {5 a \,d^{2} \sqrt {d x +c}}{8 c}\right )}{d^{2} x^{2}}-\frac {\left (3 a \,d^{2}+8 b \,c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4 c^{\frac {5}{2}}}\) | \(73\) |
Input:
int((b*x^2+a)/x^3/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(d*x+c)^(1/2)*a*(-3*d*x+2*c)/c^2/x^2-1/4*(3*a*d^2+8*b*c^2)*arctanh((d *x+c)^(1/2)/c^(1/2))/c^(5/2)
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.78 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\left [\frac {{\left (8 \, b c^{2} + 3 \, a d^{2}\right )} \sqrt {c} x^{2} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (3 \, a c d x - 2 \, a c^{2}\right )} \sqrt {d x + c}}{8 \, c^{3} x^{2}}, \frac {{\left (8 \, b c^{2} + 3 \, a d^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + {\left (3 \, a c d x - 2 \, a c^{2}\right )} \sqrt {d x + c}}{4 \, c^{3} x^{2}}\right ] \] Input:
integrate((b*x^2+a)/x^3/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
[1/8*((8*b*c^2 + 3*a*d^2)*sqrt(c)*x^2*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(3*a*c*d*x - 2*a*c^2)*sqrt(d*x + c))/(c^3*x^2), 1/4*((8*b*c^2 + 3*a*d^2)*sqrt(-c)*x^2*arctan(sqrt(-c)/sqrt(d*x + c)) + (3*a*c*d*x - 2*a *c^2)*sqrt(d*x + c))/(c^3*x^2)]
Time = 17.81 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.79 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=- \frac {a}{2 \sqrt {d} x^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1}} + \frac {a \sqrt {d}}{4 c x^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}} + \frac {3 a d^{\frac {3}{2}}}{4 c^{2} \sqrt {x} \sqrt {\frac {c}{d x} + 1}} - \frac {3 a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{4 c^{\frac {5}{2}}} + b \left (\begin {cases} \frac {2 \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{\sqrt {- c}} & \text {for}\: d \neq 0 \\\frac {\log {\left (x \right )}}{\sqrt {c}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((b*x**2+a)/x**3/(d*x+c)**(1/2),x)
Output:
-a/(2*sqrt(d)*x**(5/2)*sqrt(c/(d*x) + 1)) + a*sqrt(d)/(4*c*x**(3/2)*sqrt(c /(d*x) + 1)) + 3*a*d**(3/2)/(4*c**2*sqrt(x)*sqrt(c/(d*x) + 1)) - 3*a*d**2* asinh(sqrt(c)/(sqrt(d)*sqrt(x)))/(4*c**(5/2)) + b*Piecewise((2*atan(sqrt(c + d*x)/sqrt(-c))/sqrt(-c), Ne(d, 0)), (log(x)/sqrt(c), True))
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\frac {1}{8} \, d^{2} {\left (\frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {3}{2}} a - 5 \, \sqrt {d x + c} a c\right )}}{{\left (d x + c\right )}^{2} c^{2} - 2 \, {\left (d x + c\right )} c^{3} + c^{4}} + \frac {{\left (8 \, b c^{2} + 3 \, a d^{2}\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {5}{2}} d^{2}}\right )} \] Input:
integrate((b*x^2+a)/x^3/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
1/8*d^2*(2*(3*(d*x + c)^(3/2)*a - 5*sqrt(d*x + c)*a*c)/((d*x + c)^2*c^2 - 2*(d*x + c)*c^3 + c^4) + (8*b*c^2 + 3*a*d^2)*log((sqrt(d*x + c) - sqrt(c)) /(sqrt(d*x + c) + sqrt(c)))/(c^(5/2)*d^2))
Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\frac {\frac {{\left (8 \, b c^{2} d + 3 \, a d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{2}} + \frac {3 \, {\left (d x + c\right )}^{\frac {3}{2}} a d^{3} - 5 \, \sqrt {d x + c} a c d^{3}}{c^{2} d^{2} x^{2}}}{4 \, d} \] Input:
integrate((b*x^2+a)/x^3/(d*x+c)^(1/2),x, algorithm="giac")
Output:
1/4*((8*b*c^2*d + 3*a*d^3)*arctan(sqrt(d*x + c)/sqrt(-c))/(sqrt(-c)*c^2) + (3*(d*x + c)^(3/2)*a*d^3 - 5*sqrt(d*x + c)*a*c*d^3)/(c^2*d^2*x^2))/d
Time = 8.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\frac {3\,a\,{\left (c+d\,x\right )}^{3/2}}{4\,c^2\,x^2}-\frac {5\,a\,\sqrt {c+d\,x}}{4\,c\,x^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c+d\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,\left (8\,b\,c^2+3\,a\,d^2\right )\,1{}\mathrm {i}}{4\,c^{5/2}} \] Input:
int((a + b*x^2)/(x^3*(c + d*x)^(1/2)),x)
Output:
(atan(((c + d*x)^(1/2)*1i)/c^(1/2))*(3*a*d^2 + 8*b*c^2)*1i)/(4*c^(5/2)) - (5*a*(c + d*x)^(1/2))/(4*c*x^2) + (3*a*(c + d*x)^(3/2))/(4*c^2*x^2)
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.51 \[ \int \frac {a+b x^2}{x^3 \sqrt {c+d x}} \, dx=\frac {-4 \sqrt {d x +c}\, a \,c^{2}+6 \sqrt {d x +c}\, a c d x +3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2} x^{2}+8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b \,c^{2} x^{2}-3 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2} x^{2}-8 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b \,c^{2} x^{2}}{8 c^{3} x^{2}} \] Input:
int((b*x^2+a)/x^3/(d*x+c)^(1/2),x)
Output:
( - 4*sqrt(c + d*x)*a*c**2 + 6*sqrt(c + d*x)*a*c*d*x + 3*sqrt(c)*log(sqrt( c + d*x) - sqrt(c))*a*d**2*x**2 + 8*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*b *c**2*x**2 - 3*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*d**2*x**2 - 8*sqrt(c )*log(sqrt(c + d*x) + sqrt(c))*b*c**2*x**2)/(8*c**3*x**2)