\(\int \frac {(a+b x)^{9/2}}{x^6} \, dx\) [322]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}} \]

[Out]

-21/64*b^3*(b*x+a)^(3/2)/x^2-21/80*b^2*(b*x+a)^(5/2)/x^3-9/40*b*(b*x+a)^(7/2)/x^4-1/5*(b*x+a)^(9/2)/x^5-63/128
*b^5*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)-63/128*b^4*(b*x+a)^(1/2)/x

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {43, 65, 214} \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=-\frac {63 b^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}}-\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {(a+b x)^{9/2}}{5 x^5}-\frac {9 b (a+b x)^{7/2}}{40 x^4} \]

[In]

Int[(a + b*x)^(9/2)/x^6,x]

[Out]

(-63*b^4*Sqrt[a + b*x])/(128*x) - (21*b^3*(a + b*x)^(3/2))/(64*x^2) - (21*b^2*(a + b*x)^(5/2))/(80*x^3) - (9*b
*(a + b*x)^(7/2))/(40*x^4) - (a + b*x)^(9/2)/(5*x^5) - (63*b^5*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*Sqrt[a])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{10} (9 b) \int \frac {(a+b x)^{7/2}}{x^5} \, dx \\ & = -\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{80} \left (63 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^4} \, dx \\ & = -\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{32} \left (21 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^3} \, dx \\ & = -\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{128} \left (63 b^4\right ) \int \frac {\sqrt {a+b x}}{x^2} \, dx \\ & = -\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{256} \left (63 b^5\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = -\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}+\frac {1}{128} \left (63 b^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right ) \\ & = -\frac {63 b^4 \sqrt {a+b x}}{128 x}-\frac {21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac {21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac {9 b (a+b x)^{7/2}}{40 x^4}-\frac {(a+b x)^{9/2}}{5 x^5}-\frac {63 b^5 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 \sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=\frac {1}{640} \left (-\frac {\sqrt {a+b x} \left (128 a^4+656 a^3 b x+1368 a^2 b^2 x^2+1490 a b^3 x^3+965 b^4 x^4\right )}{x^5}-\frac {315 b^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \]

[In]

Integrate[(a + b*x)^(9/2)/x^6,x]

[Out]

(-((Sqrt[a + b*x]*(128*a^4 + 656*a^3*b*x + 1368*a^2*b^2*x^2 + 1490*a*b^3*x^3 + 965*b^4*x^4))/x^5) - (315*b^5*A
rcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a])/640

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.63

method	result	size
risch	\(-\frac {\sqrt {b x +a}\, \left (965 b^{4} x^{4}+1490 a \,b^{3} x^{3}+1368 a^{2} b^{2} x^{2}+656 a^{3} b x +128 a^{4}\right )}{640 x^{5}}-\frac {63 b^{5} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}\)	\(75\)
derivativedivides	\(2 b^{5} \left (-\frac {\frac {193 \left (b x +a \right )^{\frac {9}{2}}}{256}-\frac {237 a \left (b x +a \right )^{\frac {7}{2}}}{128}+\frac {21 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10}-\frac {147 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{128}+\frac {63 a^{4} \sqrt {b x +a}}{256}}{b^{5} x^{5}}-\frac {63 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 \sqrt {a}}\right )\)	\(88\)
default	\(2 b^{5} \left (-\frac {\frac {193 \left (b x +a \right )^{\frac {9}{2}}}{256}-\frac {237 a \left (b x +a \right )^{\frac {7}{2}}}{128}+\frac {21 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10}-\frac {147 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{128}+\frac {63 a^{4} \sqrt {b x +a}}{256}}{b^{5} x^{5}}-\frac {63 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 \sqrt {a}}\right )\)	\(88\)
pseudoelliptic	\(\frac {-315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{5} x^{5}-128 \sqrt {b x +a}\, a^{\frac {9}{2}}-656 a^{\frac {7}{2}} \sqrt {b x +a}\, b x -1368 a^{\frac {5}{2}} b^{2} x^{2} \sqrt {b x +a}-1490 a^{\frac {3}{2}} b^{3} x^{3} \sqrt {b x +a}-965 b^{4} x^{4} \sqrt {b x +a}\, \sqrt {a}}{640 x^{5} \sqrt {a}}\)	\(110\)

[In]

int((b*x+a)^(9/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/640*(b*x+a)^(1/2)*(965*b^4*x^4+1490*a*b^3*x^3+1368*a^2*b^2*x^2+656*a^3*b*x+128*a^4)/x^5-63/128*b^5*arctanh(
(b*x+a)^(1/2)/a^(1/2))/a^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=\left [\frac {315 \, \sqrt {a} b^{5} x^{5} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (965 \, a b^{4} x^{4} + 1490 \, a^{2} b^{3} x^{3} + 1368 \, a^{3} b^{2} x^{2} + 656 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x + a}}{1280 \, a x^{5}}, \frac {315 \, \sqrt {-a} b^{5} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (965 \, a b^{4} x^{4} + 1490 \, a^{2} b^{3} x^{3} + 1368 \, a^{3} b^{2} x^{2} + 656 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x + a}}{640 \, a x^{5}}\right ] \]

[In]

integrate((b*x+a)^(9/2)/x^6,x, algorithm="fricas")

[Out]

[1/1280*(315*sqrt(a)*b^5*x^5*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*(965*a*b^4*x^4 + 1490*a^2*b^3*x^
3 + 1368*a^3*b^2*x^2 + 656*a^4*b*x + 128*a^5)*sqrt(b*x + a))/(a*x^5), 1/640*(315*sqrt(-a)*b^5*x^5*arctan(sqrt(
b*x + a)*sqrt(-a)/a) - (965*a*b^4*x^4 + 1490*a^2*b^3*x^3 + 1368*a^3*b^2*x^2 + 656*a^4*b*x + 128*a^5)*sqrt(b*x
+ a))/(a*x^5)]

Sympy [A] (verification not implemented)

Time = 14.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=- \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{5 x^{\frac {9}{2}}} - \frac {41 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{40 x^{\frac {7}{2}}} - \frac {171 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{80 x^{\frac {5}{2}}} - \frac {149 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}}{64 x^{\frac {3}{2}}} - \frac {193 b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{128 \sqrt {x}} - \frac {63 b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{128 \sqrt {a}} \]

[In]

integrate((b*x+a)**(9/2)/x**6,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x**(9/2)) - 41*a**3*b**(3/2)*sqrt(a/(b*x) + 1)/(40*x**(7/2)) - 171*a**2*b**
(5/2)*sqrt(a/(b*x) + 1)/(80*x**(5/2)) - 149*a*b**(7/2)*sqrt(a/(b*x) + 1)/(64*x**(3/2)) - 193*b**(9/2)*sqrt(a/(
b*x) + 1)/(128*sqrt(x)) - 63*b**5*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(128*sqrt(a))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=\frac {63 \, b^{5} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{256 \, \sqrt {a}} - \frac {965 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{5} - 2370 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{5} + 2688 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{5} - 1470 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{5} + 315 \, \sqrt {b x + a} a^{4} b^{5}}{640 \, {\left ({\left (b x + a\right )}^{5} - 5 \, {\left (b x + a\right )}^{4} a + 10 \, {\left (b x + a\right )}^{3} a^{2} - 10 \, {\left (b x + a\right )}^{2} a^{3} + 5 \, {\left (b x + a\right )} a^{4} - a^{5}\right )}} \]

[In]

integrate((b*x+a)^(9/2)/x^6,x, algorithm="maxima")

[Out]

63/256*b^5*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/sqrt(a) - 1/640*(965*(b*x + a)^(9/2)*b^5 -
2370*(b*x + a)^(7/2)*a*b^5 + 2688*(b*x + a)^(5/2)*a^2*b^5 - 1470*(b*x + a)^(3/2)*a^3*b^5 + 315*sqrt(b*x + a)*
a^4*b^5)/((b*x + a)^5 - 5*(b*x + a)^4*a + 10*(b*x + a)^3*a^2 - 10*(b*x + a)^2*a^3 + 5*(b*x + a)*a^4 - a^5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=\frac {\frac {315 \, b^{6} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {965 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{6} - 2370 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{6} + 2688 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{6} - 1470 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{6} + 315 \, \sqrt {b x + a} a^{4} b^{6}}{b^{5} x^{5}}}{640 \, b} \]

[In]

integrate((b*x+a)^(9/2)/x^6,x, algorithm="giac")

[Out]

1/640*(315*b^6*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - (965*(b*x + a)^(9/2)*b^6 - 2370*(b*x + a)^(7/2)*a*b^6
+ 2688*(b*x + a)^(5/2)*a^2*b^6 - 1470*(b*x + a)^(3/2)*a^3*b^6 + 315*sqrt(b*x + a)*a^4*b^6)/(b^5*x^5))/b

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b x)^{9/2}}{x^6} \, dx=\frac {147\,a^3\,{\left (a+b\,x\right )}^{3/2}}{64\,x^5}-\frac {63\,a^4\,\sqrt {a+b\,x}}{128\,x^5}-\frac {193\,{\left (a+b\,x\right )}^{9/2}}{128\,x^5}-\frac {21\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5\,x^5}+\frac {237\,a\,{\left (a+b\,x\right )}^{7/2}}{64\,x^5}+\frac {b^5\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{128\,\sqrt {a}} \]

[In]

int((a + b*x)^(9/2)/x^6,x)

[Out]

(147*a^3*(a + b*x)^(3/2))/(64*x^5) - (63*a^4*(a + b*x)^(1/2))/(128*x^5) - (193*(a + b*x)^(9/2))/(128*x^5) - (2
1*a^2*(a + b*x)^(5/2))/(5*x^5) + (b^5*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*63i)/(128*a^(1/2)) + (237*a*(a + b*x)
^(7/2))/(64*x^5)