\(\int \frac {(a x^2+b x^3)^{3/2}}{x^9} \, dx\) [251]

Optimal result

Integrand size = 19, antiderivative size = 165 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=-\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {3 b^5 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}} \]

[Out]

-1/5*(b*x^3+a*x^2)^(3/2)/x^8+3/128*b^5*arctanh(x*a^(1/2)/(b*x^3+a*x^2)^(1/2))/a^(7/2)-3/40*b*(b*x^3+a*x^2)^(1/
2)/x^5-1/80*b^2*(b*x^3+a*x^2)^(1/2)/a/x^4+1/64*b^3*(b*x^3+a*x^2)^(1/2)/a^2/x^3-3/128*b^4*(b*x^3+a*x^2)^(1/2)/a
^3/x^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2045, 2050, 2033, 212} \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\frac {3 b^5 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5} \]

[In]

Int[(a*x^2 + b*x^3)^(3/2)/x^9,x]

[Out]

(-3*b*Sqrt[a*x^2 + b*x^3])/(40*x^5) - (b^2*Sqrt[a*x^2 + b*x^3])/(80*a*x^4) + (b^3*Sqrt[a*x^2 + b*x^3])/(64*a^2
*x^3) - (3*b^4*Sqrt[a*x^2 + b*x^3])/(128*a^3*x^2) - (a*x^2 + b*x^3)^(3/2)/(5*x^8) + (3*b^5*ArcTanh[(Sqrt[a]*x)
/Sqrt[a*x^2 + b*x^3]])/(128*a^(7/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2033

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rule 2045

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] - Dist[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] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2050

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] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {1}{10} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^6} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {1}{80} \left (3 b^2\right ) \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {b^3 \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{32 a} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {\left (3 b^4\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{128 a^2} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {\left (3 b^5\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{256 a^3} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {\left (3 b^5\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{128 a^3} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {3 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\frac {\sqrt {x^2 (a+b x)} \left (-\sqrt {a} \sqrt {a+b x} \left (128 a^4+176 a^3 b x+8 a^2 b^2 x^2-10 a b^3 x^3+15 b^4 x^4\right )+15 b^5 x^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{640 a^{7/2} x^6 \sqrt {a+b x}} \]

[In]

Integrate[(a*x^2 + b*x^3)^(3/2)/x^9,x]

[Out]

(Sqrt[x^2*(a + b*x)]*(-(Sqrt[a]*Sqrt[a + b*x]*(128*a^4 + 176*a^3*b*x + 8*a^2*b^2*x^2 - 10*a*b^3*x^3 + 15*b^4*x
^4)) + 15*b^5*x^5*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(640*a^(7/2)*x^6*Sqrt[a + b*x])

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62

method	result	size
risch	\(-\frac {\left (15 b^{4} x^{4}-10 a \,b^{3} x^{3}+8 a^{2} b^{2} x^{2}+176 a^{3} b x +128 a^{4}\right ) \sqrt {x^{2} \left (b x +a \right )}}{640 x^{6} a^{3}}+\frac {3 b^{5} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{128 a^{\frac {7}{2}} x \sqrt {b x +a}}\)	\(103\)
default	\(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (15 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {7}{2}}-70 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {9}{2}}+128 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {11}{2}}-15 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{3} x^{5} b^{5}+70 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {13}{2}}-15 \sqrt {b x +a}\, a^{\frac {15}{2}}\right )}{640 x^{8} \left (b x +a \right )^{\frac {3}{2}} a^{\frac {13}{2}}}\)	\(113\)
pseudoelliptic	\(-\frac {\frac {693 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{8} x^{8}}{256}+\sqrt {b x +a}\, \left (-\frac {693 \sqrt {a}\, b^{7} x^{7}}{256}+\frac {231 a^{\frac {3}{2}} b^{6} x^{6}}{128}-\frac {231 a^{\frac {5}{2}} b^{5} x^{5}}{160}+\frac {99 a^{\frac {7}{2}} b^{4} x^{4}}{80}-\frac {11 a^{\frac {9}{2}} b^{3} x^{3}}{10}+a^{\frac {11}{2}} b^{2} x^{2}+68 a^{\frac {13}{2}} b x +56 a^{\frac {15}{2}}\right )}{448 a^{\frac {13}{2}} x^{8}}\)	\(116\)

[In]

int((b*x^3+a*x^2)^(3/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/640*(15*b^4*x^4-10*a*b^3*x^3+8*a^2*b^2*x^2+176*a^3*b*x+128*a^4)/x^6/a^3*(x^2*(b*x+a))^(1/2)+3/128*b^5/a^(7/
2)*arctanh((b*x+a)^(1/2)/a^(1/2))/x/(b*x+a)^(1/2)*(x^2*(b*x+a))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\left [\frac {15 \, \sqrt {a} b^{5} x^{6} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{1280 \, a^{4} x^{6}}, -\frac {15 \, \sqrt {-a} b^{5} x^{6} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{640 \, a^{4} x^{6}}\right ] \]

[In]

integrate((b*x^3+a*x^2)^(3/2)/x^9,x, algorithm="fricas")

[Out]

[1/1280*(15*sqrt(a)*b^5*x^6*log((b*x^2 + 2*a*x + 2*sqrt(b*x^3 + a*x^2)*sqrt(a))/x^2) - 2*(15*a*b^4*x^4 - 10*a^
2*b^3*x^3 + 8*a^3*b^2*x^2 + 176*a^4*b*x + 128*a^5)*sqrt(b*x^3 + a*x^2))/(a^4*x^6), -1/640*(15*sqrt(-a)*b^5*x^6
*arctan(sqrt(b*x^3 + a*x^2)*sqrt(-a)/(a*x)) + (15*a*b^4*x^4 - 10*a^2*b^3*x^3 + 8*a^3*b^2*x^2 + 176*a^4*b*x + 1
28*a^5)*sqrt(b*x^3 + a*x^2))/(a^4*x^6)]

Sympy [F]

\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{9}}\, dx \]

[In]

integrate((b*x**3+a*x**2)**(3/2)/x**9,x)

[Out]

Integral((x**2*(a + b*x))**(3/2)/x**9, x)

Maxima [F]

\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{9}} \,d x } \]

[In]

integrate((b*x^3+a*x^2)^(3/2)/x^9,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a*x^2)^(3/2)/x^9, x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=-\frac {\frac {15 \, b^{6} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{6} \mathrm {sgn}\left (x\right ) - 70 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{6} \mathrm {sgn}\left (x\right ) + 128 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{6} \mathrm {sgn}\left (x\right ) + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{6} \mathrm {sgn}\left (x\right ) - 15 \, \sqrt {b x + a} a^{4} b^{6} \mathrm {sgn}\left (x\right )}{a^{3} b^{5} x^{5}}}{640 \, b} \]

[In]

integrate((b*x^3+a*x^2)^(3/2)/x^9,x, algorithm="giac")

[Out]

-1/640*(15*b^6*arctan(sqrt(b*x + a)/sqrt(-a))*sgn(x)/(sqrt(-a)*a^3) + (15*(b*x + a)^(9/2)*b^6*sgn(x) - 70*(b*x
+ a)^(7/2)*a*b^6*sgn(x) + 128*(b*x + a)^(5/2)*a^2*b^6*sgn(x) + 70*(b*x + a)^(3/2)*a^3*b^6*sgn(x) - 15*sqrt(b*
x + a)*a^4*b^6*sgn(x))/(a^3*b^5*x^5))/b

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^9} \,d x \]

[In]

int((a*x^2 + b*x^3)^(3/2)/x^9,x)

[Out]

int((a*x^2 + b*x^3)^(3/2)/x^9, x)