\(\int \frac {(a x^2+b x^3)^{3/2}}{x^8} \, dx\) [229]

Optimal result

Integrand size = 19, antiderivative size = 137 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=-\frac {b \sqrt {a x^2+b x^3}}{8 x^4}-\frac {b^2 \sqrt {a x^2+b x^3}}{32 a x^3}+\frac {3 b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac {3 b^4 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{64 a^{5/2}} \] Output:

-1/8*b*(b*x^3+a*x^2)^(1/2)/x^4-1/32*b^2*(b*x^3+a*x^2)^(1/2)/a/x^3+3/64*b^3 
*(b*x^3+a*x^2)^(1/2)/a^2/x^2-1/4*(b*x^3+a*x^2)^(3/2)/x^7-3/64*b^4*arctanh( 
a^(1/2)*x/(b*x^3+a*x^2)^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=-\frac {\sqrt {x^2 (a+b x)} \left (\sqrt {a} \sqrt {a+b x} \left (16 a^3+24 a^2 b x+2 a b^2 x^2-3 b^3 x^3\right )+3 b^4 x^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{64 a^{5/2} x^5 \sqrt {a+b x}} \] Input:

Integrate[(a*x^2 + b*x^3)^(3/2)/x^8,x]
 

Output:

-1/64*(Sqrt[x^2*(a + b*x)]*(Sqrt[a]*Sqrt[a + b*x]*(16*a^3 + 24*a^2*b*x + 2 
*a*b^2*x^2 - 3*b^3*x^3) + 3*b^4*x^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(a^(5 
/2)*x^5*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1926, 1926, 1931, 1931, 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.

\(\displaystyle \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx\)

\(\Big \downarrow \) 1926

\(\displaystyle \frac {3}{8} b \int \frac {\sqrt {b x^3+a x^2}}{x^5}dx-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

\(\Big \downarrow \) 1926

\(\displaystyle \frac {3}{8} b \left (\frac {1}{6} b \int \frac {1}{x^2 \sqrt {b x^3+a x^2}}dx-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x \sqrt {b x^3+a x^2}}dx}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{\sqrt {b x^3+a x^2}}dx}{2 a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

\(\Big \downarrow \) 1914

\(\displaystyle \frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \int \frac {1}{1-\frac {a x^2}{b x^3+a x^2}}d\frac {x}{\sqrt {b x^3+a x^2}}}{a}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{a^{3/2}}-\frac {\sqrt {a x^2+b x^3}}{a x^2}\right )}{4 a}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}\right )-\frac {\sqrt {a x^2+b x^3}}{3 x^4}\right )-\frac {\left (a x^2+b x^3\right )^{3/2}}{4 x^7}\)

Input:

Int[(a*x^2 + b*x^3)^(3/2)/x^8,x]
 

Output:

-1/4*(a*x^2 + b*x^3)^(3/2)/x^7 + (3*b*(-1/3*Sqrt[a*x^2 + b*x^3]/x^4 + (b*( 
-1/2*Sqrt[a*x^2 + b*x^3]/(a*x^3) - (3*b*(-(Sqrt[a*x^2 + b*x^3]/(a*x^2)) + 
(b*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/a^(3/2)))/(4*a)))/6))/8
 

Defintions of rubi rules used

rule 219

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])
 

rule 1914

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]
 

rule 1926

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]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] 
 &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ 
m + j*p + 1, 0]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (-3 b^{3} x^{3}+2 a \,b^{2} x^{2}+24 a^{2} b x +16 a^{3}\right ) \sqrt {x^{2} \left (b x +a \right )}}{64 x^{5} a^{2}}-\frac {3 b^{4} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{64 a^{\frac {5}{2}} x \sqrt {b x +a}}\)	\(92\)
default	\(\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (3 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {5}{2}}-11 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {7}{2}}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{2} b^{4} x^{4}-11 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}}+3 a^{\frac {11}{2}} \sqrt {b x +a}\right )}{64 x^{7} \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}}}\)	\(101\)
pseudoelliptic	\(-\frac {5 \left (-\frac {63 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) x^{7} b^{7}}{1280}+\sqrt {b x +a}\, \left (\frac {63 \sqrt {a}\, b^{6} x^{6}}{1280}-\frac {21 a^{\frac {3}{2}} b^{5} x^{5}}{640}+\frac {21 a^{\frac {5}{2}} b^{4} x^{4}}{800}-\frac {9 a^{\frac {7}{2}} b^{3} x^{3}}{400}+\frac {a^{\frac {9}{2}} b^{2} x^{2}}{50}+a^{\frac {11}{2}} b x +\frac {4 a^{\frac {13}{2}}}{5}\right )\right )}{28 a^{\frac {11}{2}} x^{7}}\)	\(105\)

Input:

int((b*x^3+a*x^2)^(3/2)/x^8,x,method=_RETURNVERBOSE)
 

Output:

-1/64*(-3*b^3*x^3+2*a*b^2*x^2+24*a^2*b*x+16*a^3)/x^5/a^2*(x^2*(b*x+a))^(1/ 
2)-3/64/a^(5/2)*b^4*arctanh((b*x+a)^(1/2)/a^(1/2))*(x^2*(b*x+a))^(1/2)/x/( 
b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\left [\frac {3 \, \sqrt {a} b^{4} x^{5} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, {\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}, \frac {3 \, \sqrt {-a} b^{4} x^{5} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{b x^{2} + a x}\right ) + {\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x^{3} + a x^{2}}}{64 \, a^{3} x^{5}}\right ] \] Input:

integrate((b*x^3+a*x^2)^(3/2)/x^8,x, algorithm="fricas")
 

Output:

[1/128*(3*sqrt(a)*b^4*x^5*log((b*x^2 + 2*a*x - 2*sqrt(b*x^3 + a*x^2)*sqrt( 
a))/x^2) + 2*(3*a*b^3*x^3 - 2*a^2*b^2*x^2 - 24*a^3*b*x - 16*a^4)*sqrt(b*x^ 
3 + a*x^2))/(a^3*x^5), 1/64*(3*sqrt(-a)*b^4*x^5*arctan(sqrt(b*x^3 + a*x^2) 
*sqrt(-a)/(b*x^2 + a*x)) + (3*a*b^3*x^3 - 2*a^2*b^2*x^2 - 24*a^3*b*x - 16* 
a^4)*sqrt(b*x^3 + a*x^2))/(a^3*x^5)]
 

Sympy [F]

\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{8}}\, dx \] Input:

integrate((b*x**3+a*x**2)**(3/2)/x**8,x)
 

Output:

Integral((x**2*(a + b*x))**(3/2)/x**8, x)
 

Maxima [F]

\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:

integrate((b*x^3+a*x^2)^(3/2)/x^8,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a*x^2)^(3/2)/x^8, x)
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\frac {\frac {3 \, b^{5} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {7}{2}} b^{5} \mathrm {sgn}\left (x\right ) - 11 \, {\left (b x + a\right )}^{\frac {5}{2}} a b^{5} \mathrm {sgn}\left (x\right ) - 11 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} b^{5} \mathrm {sgn}\left (x\right ) + 3 \, \sqrt {b x + a} a^{3} b^{5} \mathrm {sgn}\left (x\right )}{a^{2} b^{4} x^{4}}}{64 \, b} \] Input:

integrate((b*x^3+a*x^2)^(3/2)/x^8,x, algorithm="giac")
 

Output:

1/64*(3*b^5*arctan(sqrt(b*x + a)/sqrt(-a))*sgn(x)/(sqrt(-a)*a^2) + (3*(b*x 
 + a)^(7/2)*b^5*sgn(x) - 11*(b*x + a)^(5/2)*a*b^5*sgn(x) - 11*(b*x + a)^(3 
/2)*a^2*b^5*sgn(x) + 3*sqrt(b*x + a)*a^3*b^5*sgn(x))/(a^2*b^4*x^4))/b
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^8} \,d x \] Input:

int((a*x^2 + b*x^3)^(3/2)/x^8,x)
 

Output:

int((a*x^2 + b*x^3)^(3/2)/x^8, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx=\frac {-32 \sqrt {b x +a}\, a^{4}-48 \sqrt {b x +a}\, a^{3} b x -4 \sqrt {b x +a}\, a^{2} b^{2} x^{2}+6 \sqrt {b x +a}\, a \,b^{3} x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{4} x^{4}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{4} x^{4}}{128 a^{3} x^{4}} \] Input:

int((b*x^3+a*x^2)^(3/2)/x^8,x)
 

Output:

( - 32*sqrt(a + b*x)*a**4 - 48*sqrt(a + b*x)*a**3*b*x - 4*sqrt(a + b*x)*a* 
*2*b**2*x**2 + 6*sqrt(a + b*x)*a*b**3*x**3 + 3*sqrt(a)*log(sqrt(a + b*x) - 
 sqrt(a))*b**4*x**4 - 3*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b**4*x**4)/(1 
28*a**3*x**4)