Integrand size = 19, antiderivative size = 73 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\frac {3 b \sqrt {a x^2+b x^3}}{x}-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}-3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right ) \] Output:
3*b*(b*x^3+a*x^2)^(1/2)/x-(b*x^3+a*x^2)^(3/2)/x^4-3*a^(1/2)*b*arctanh(a^(1 /2)*x/(b*x^3+a*x^2)^(1/2))
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=-\frac {\sqrt {a+b x} \left ((a-2 b x) \sqrt {a+b x}+3 \sqrt {a} b x \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{\sqrt {x^2 (a+b x)}} \] Input:
Integrate[(a*x^2 + b*x^3)^(3/2)/x^5,x]
Output:
-((Sqrt[a + b*x]*((a - 2*b*x)*Sqrt[a + b*x] + 3*Sqrt[a]*b*x*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/Sqrt[x^2*(a + b*x)])
Time = 0.38 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1926, 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.
\(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx\) |
\(\Big \downarrow \) 1926 |
\(\displaystyle \frac {3}{2} b \int \frac {\sqrt {b x^3+a x^2}}{x^2}dx-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {3}{2} b \left (a \int \frac {1}{\sqrt {b x^3+a x^2}}dx+\frac {2 \sqrt {a x^2+b x^3}}{x}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 1914 |
\(\displaystyle \frac {3}{2} b \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 a \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{2} b \left (\frac {2 \sqrt {a x^2+b x^3}}{x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{x^4}\) |
Input:
Int[(a*x^2 + b*x^3)^(3/2)/x^5,x]
Output:
-((a*x^2 + b*x^3)^(3/2)/x^4) + (3*b*((2*Sqrt[a*x^2 + b*x^3])/x - 2*Sqrt[a] *ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]]))/2
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 + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
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 = 0.51 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {a \sqrt {x^{2} \left (b x +a \right )}}{x^{2}}+\frac {b \left (4 \sqrt {b x +a}-6 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right ) \sqrt {x^{2} \left (b x +a \right )}}{2 x \sqrt {b x +a}}\) | \(70\) |
default | \(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-2 b x \sqrt {b x +a}\, \sqrt {a}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a b x +a^{\frac {3}{2}} \sqrt {b x +a}\right )}{x^{4} \left (b x +a \right )^{\frac {3}{2}} \sqrt {a}}\) | \(72\) |
pseudoelliptic | \(\frac {-\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{4} x^{4}}{64}+\frac {3 \sqrt {b x +a}\, \left (\sqrt {a}\, b^{3} x^{3}-\frac {2 a^{\frac {3}{2}} b^{2} x^{2}}{3}-8 a^{\frac {5}{2}} b x -\frac {16 a^{\frac {7}{2}}}{3}\right )}{64}}{a^{\frac {5}{2}} x^{4}}\) | \(72\) |
Input:
int((b*x^3+a*x^2)^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-a/x^2*(x^2*(b*x+a))^(1/2)+1/2*b*(4*(b*x+a)^(1/2)-6*a^(1/2)*arctanh((b*x+a )^(1/2)/a^(1/2)))*(x^2*(b*x+a))^(1/2)/x/(b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\left [\frac {3 \, \sqrt {a} b x^{2} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{2 \, x^{2}}, \frac {3 \, \sqrt {-a} b x^{2} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (2 \, b x - a\right )}}{x^{2}}\right ] \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^5,x, algorithm="fricas")
Output:
[1/2*(3*sqrt(a)*b*x^2*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)*(2*b*x - a))/x^2, (3*sqrt(-a)*b*x^2*arctan(sq rt(b*x^3 + a*x^2)*sqrt(-a)/(b*x^2 + a*x)) + sqrt(b*x^3 + a*x^2)*(2*b*x - a ))/x^2]
\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:
integrate((b*x**3+a*x**2)**(3/2)/x**5,x)
Output:
Integral((x**2*(a + b*x))**(3/2)/x**5, x)
\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^5,x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(3/2)/x^5, x)
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx={\left (\frac {3 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + 2 \, \sqrt {b x + a} \mathrm {sgn}\left (x\right ) - \frac {\sqrt {b x + a} a \mathrm {sgn}\left (x\right )}{b x}\right )} b \] Input:
integrate((b*x^3+a*x^2)^(3/2)/x^5,x, algorithm="giac")
Output:
(3*a*arctan(sqrt(b*x + a)/sqrt(-a))*sgn(x)/sqrt(-a) + 2*sqrt(b*x + a)*sgn( x) - sqrt(b*x + a)*a*sgn(x)/(b*x))*b
Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^5} \,d x \] Input:
int((a*x^2 + b*x^3)^(3/2)/x^5,x)
Output:
int((a*x^2 + b*x^3)^(3/2)/x^5, x)
Time = 0.22 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^5} \, dx=\frac {-2 \sqrt {b x +a}\, a +4 \sqrt {b x +a}\, b x +3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b x -3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b x}{2 x} \] Input:
int((b*x^3+a*x^2)^(3/2)/x^5,x)
Output:
( - 2*sqrt(a + b*x)*a + 4*sqrt(a + b*x)*b*x + 3*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b*x - 3*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b*x)/(2*x)