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

Optimal. Leaf size=119 \[ -\frac{21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac{21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac{63 b^4 \sqrt{a+b x}}{128 x}-\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 \sqrt{a}}-\frac{9 b (a+b x)^{7/2}}{40 x^4}-\frac{(a+b x)^{9/2}}{5 x^5} \]

[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])

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Rubi [A]  time = 0.0385725, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {47, 63, 208} \[ -\frac{21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac{21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac{63 b^4 \sqrt{a+b x}}{128 x}-\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 \sqrt{a}}-\frac{9 b (a+b x)^{7/2}}{40 x^4}-\frac{(a+b x)^{9/2}}{5 x^5} \]

Antiderivative was successfully verified.

[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 47

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},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 208

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

Rubi steps

\begin{align*} \int \frac{(a+b x)^{9/2}}{x^6} \, dx &=-\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 ) \operatorname{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]  time = 0.130866, size = 101, normalized size = 0.85 \[ -\frac{2024 a^3 b^2 x^2+2858 a^2 b^3 x^3+784 a^4 b x+128 a^5+2455 a b^4 x^4+315 b^5 x^5 \sqrt{\frac{b x}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x}{a}+1}\right )+965 b^5 x^5}{640 x^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2)/x^6,x]

[Out]

-(128*a^5 + 784*a^4*b*x + 2024*a^3*b^2*x^2 + 2858*a^2*b^3*x^3 + 2455*a*b^4*x^4 + 965*b^5*x^5 + 315*b^5*x^5*Sqr
t[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]])/(640*x^5*Sqrt[a + b*x])

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Maple [A]  time = 0.013, size = 87, normalized size = 0.7 \begin{align*} 2\,{b}^{5} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ( -{\frac{193\, \left ( bx+a \right ) ^{9/2}}{256}}+{\frac{237\,a \left ( bx+a \right ) ^{7/2}}{128}}-{\frac{21\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{10}}+{\frac{147\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{128}}-{\frac{63\,{a}^{4}\sqrt{bx+a}}{256}} \right ) }-{\frac{63}{256\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^5*((-193/256*(b*x+a)^(9/2)+237/128*a*(b*x+a)^(7/2)-21/10*a^2*(b*x+a)^(5/2)+147/128*a^3*(b*x+a)^(3/2)-63/25
6*a^4*(b*x+a)^(1/2))/b^5/x^5-63/256*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.6636, size = 470, normalized size = 3.95 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]  time = 13.3973, size = 158, normalized size = 1.33 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.20668, size = 147, normalized size = 1.24 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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