Integrand size = 19, antiderivative size = 74 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\frac {2 a \sqrt {a x^2+b x^3}}{x}+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right ) \] Output:
2*a*(b*x^3+a*x^2)^(1/2)/x+2/3*(b*x^3+a*x^2)^(3/2)/x^3-2*a^(3/2)*arctanh(a^ (1/2)*x/(b*x^3+a*x^2)^(1/2))
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\frac {2 x \sqrt {a+b x} \left (\sqrt {a+b x} (4 a+b x)-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{3 \sqrt {x^2 (a+b x)}} \] Input:
Integrate[(a*x^2 + b*x^3)^(3/2)/x^4,x]
Output:
(2*x*Sqrt[a + b*x]*(Sqrt[a + b*x]*(4*a + b*x) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(3*Sqrt[x^2*(a + b*x)])
Time = 0.38 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1927, 1927, 1914, 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.
\(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle a \int \frac {\sqrt {b x^3+a x^2}}{x^2}dx+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle a \left (a \int \frac {1}{\sqrt {b x^3+a x^2}}dx+\frac {2 \sqrt {a x^2+b x^3}}{x}\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 1914 |
\(\displaystyle a \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 a \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\right )+\frac {2 \left (a x^2+b x^3\right )^{3/2}}{3 x^3}\) |
Input:
Int[(a*x^2 + b*x^3)^(3/2)/x^4,x]
Output:
(2*(a*x^2 + b*x^3)^(3/2))/(3*x^3) + a*((2*Sqrt[a*x^2 + b*x^3])/x - 2*Sqrt[ a]*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])
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[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[2/(2 - n) Subst[Int[1/(1 - a*x^2), x], x, x/Sqrt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Time = 0.48 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {2 \left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-3 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\left (b x +a \right )^{\frac {3}{2}}+3 a \sqrt {b x +a}\right )}{3 x^{3} \left (b x +a \right )^{\frac {3}{2}}}\) | \(61\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{3} x^{3}-\left (\sqrt {a}\, b^{2} x^{2}+\frac {14 a^{\frac {3}{2}} b x}{3}+\frac {8 a^{\frac {5}{2}}}{3}\right ) \sqrt {b x +a}}{8 a^{\frac {3}{2}} x^{3}}\) | \(61\) |
Input:
int((b*x^3+a*x^2)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
2/3*(b*x^3+a*x^2)^(3/2)*(-3*a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2))+(b*x+a) ^(3/2)+3*a*(b*x+a)^(1/2))/x^3/(b*x+a)^(3/2)
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} x \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x + 4 \, a\right )}}{3 \, x}, \frac {2 \, {\left (3 \, \sqrt {-a} a x \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (b x + 4 \, a\right )}\right )}}{3 \, x}\right ] \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^4,x, algorithm="fricas")
Output:
[1/3*(3*a^(3/2)*x*log((b*x^2 + 2*a*x - 2*sqrt(b*x^3 + a*x^2)*sqrt(a))/x^2) + 2*sqrt(b*x^3 + a*x^2)*(b*x + 4*a))/x, 2/3*(3*sqrt(-a)*a*x*arctan(sqrt(b *x^3 + a*x^2)*sqrt(-a)/(b*x^2 + a*x)) + sqrt(b*x^3 + a*x^2)*(b*x + 4*a))/x ]
\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:
integrate((b*x**3+a*x**2)**(3/2)/x**4,x)
Output:
Integral((x**2*(a + b*x))**(3/2)/x**4, x)
\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^4,x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(3/2)/x^4, x)
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) + 2 \, \sqrt {b x + a} a \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (3 \, a^{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 4 \, \sqrt {-a} a^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-a}} \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^4,x, algorithm="giac")
Output:
2*a^2*arctan(sqrt(b*x + a)/sqrt(-a))*sgn(x)/sqrt(-a) + 2/3*(b*x + a)^(3/2) *sgn(x) + 2*sqrt(b*x + a)*a*sgn(x) - 2/3*(3*a^2*arctan(sqrt(a)/sqrt(-a)) + 4*sqrt(-a)*a^(3/2))*sgn(x)/sqrt(-a)
Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^4} \,d x \] Input:
int((a*x^2 + b*x^3)^(3/2)/x^4,x)
Output:
int((a*x^2 + b*x^3)^(3/2)/x^4, x)
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^4} \, dx=\frac {8 \sqrt {b x +a}\, a}{3}+\frac {2 \sqrt {b x +a}\, b x}{3}+\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a -\sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \] Input:
int((b*x^3+a*x^2)^(3/2)/x^4,x)
Output:
(8*sqrt(a + b*x)*a + 2*sqrt(a + b*x)*b*x + 3*sqrt(a)*log(sqrt(a + b*x) - s qrt(a))*a - 3*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a)/3