3.324 \(\int \frac{(a+b x)^{9/2}}{x^8} \, dx\)

Optimal. Leaf size=163 \[ \frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7} \]

[Out]

(-3*b^4*Sqrt[a + b*x])/(128*x^3) - (3*b^5*Sqrt[a + b*x])/(512*a*x^2) + (9*b^6*Sqrt[a + b*x])/(1024*a^2*x) - (3
*b^3*(a + b*x)^(3/2))/(64*x^4) - (3*b^2*(a + b*x)^(5/2))/(40*x^5) - (3*b*(a + b*x)^(7/2))/(28*x^6) - (a + b*x)
^(9/2)/(7*x^7) - (9*b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))

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Rubi [A]  time = 0.0675368, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 51, 63, 208} \[ \frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(9/2)/x^8,x]

[Out]

(-3*b^4*Sqrt[a + b*x])/(128*x^3) - (3*b^5*Sqrt[a + b*x])/(512*a*x^2) + (9*b^6*Sqrt[a + b*x])/(1024*a^2*x) - (3
*b^3*(a + b*x)^(3/2))/(64*x^4) - (3*b^2*(a + b*x)^(5/2))/(40*x^5) - (3*b*(a + b*x)^(7/2))/(28*x^6) - (a + b*x)
^(9/2)/(7*x^7) - (9*b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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^8} \, dx &=-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{14} (9 b) \int \frac{(a+b x)^{7/2}}{x^7} \, dx\\ &=-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{8} \left (3 b^2\right ) \int \frac{(a+b x)^{5/2}}{x^6} \, dx\\ &=-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{16} \left (3 b^3\right ) \int \frac{(a+b x)^{3/2}}{x^5} \, dx\\ &=-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{128} \left (9 b^4\right ) \int \frac{\sqrt{a+b x}}{x^4} \, dx\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{1}{256} \left (3 b^5\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{\left (9 b^6\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{1024 a}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{\left (9 b^7\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2048 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}+\frac{\left (9 b^6\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{1024 a^2}\\ &=-\frac{3 b^4 \sqrt{a+b x}}{128 x^3}-\frac{3 b^5 \sqrt{a+b x}}{512 a x^2}+\frac{9 b^6 \sqrt{a+b x}}{1024 a^2 x}-\frac{3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac{3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac{3 b (a+b x)^{7/2}}{28 x^6}-\frac{(a+b x)^{9/2}}{7 x^7}-\frac{9 b^7 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0187144, size = 35, normalized size = 0.21 \[ \frac{2 b^7 (a+b x)^{11/2} \, _2F_1\left (\frac{11}{2},8;\frac{13}{2};\frac{b x}{a}+1\right )}{11 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2)/x^8,x]

[Out]

(2*b^7*(a + b*x)^(11/2)*Hypergeometric2F1[11/2, 8, 13/2, 1 + (b*x)/a])/(11*a^8)

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Maple [A]  time = 0.011, size = 111, normalized size = 0.7 \begin{align*} 2\,{b}^{7} \left ({\frac{1}{{b}^{7}{x}^{7}} \left ({\frac{9\, \left ( bx+a \right ) ^{13/2}}{2048\,{a}^{2}}}-{\frac{15\, \left ( bx+a \right ) ^{11/2}}{512\,a}}-{\frac{1199\, \left ( bx+a \right ) ^{9/2}}{10240}}+{\frac{9\,a \left ( bx+a \right ) ^{7/2}}{70}}-{\frac{849\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{10240}}+{\frac{15\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{512}}-{\frac{9\,{a}^{4}\sqrt{bx+a}}{2048}} \right ) }-{\frac{9}{2048\,{a}^{5/2}}{\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^8,x)

[Out]

2*b^7*((9/2048/a^2*(b*x+a)^(13/2)-15/512/a*(b*x+a)^(11/2)-1199/10240*(b*x+a)^(9/2)+9/70*a*(b*x+a)^(7/2)-849/10
240*a^2*(b*x+a)^(5/2)+15/512*a^3*(b*x+a)^(3/2)-9/2048*a^4*(b*x+a)^(1/2))/b^7/x^7-9/2048*arctanh((b*x+a)^(1/2)/
a^(1/2))/a^(5/2))

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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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71013, size = 595, normalized size = 3.65 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{7} x^{7} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt{b x + a}}{71680 \, a^{3} x^{7}}, \frac{315 \, \sqrt{-a} b^{7} x^{7} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt{b x + a}}{35840 \, a^{3} x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^8,x, algorithm="fricas")

[Out]

[1/71680*(315*sqrt(a)*b^7*x^7*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(315*a*b^6*x^6 - 210*a^2*b^5*x^
5 - 14168*a^3*b^4*x^4 - 39056*a^4*b^3*x^3 - 44928*a^5*b^2*x^2 - 24320*a^6*b*x - 5120*a^7)*sqrt(b*x + a))/(a^3*
x^7), 1/35840*(315*sqrt(-a)*b^7*x^7*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (315*a*b^6*x^6 - 210*a^2*b^5*x^5 - 1416
8*a^3*b^4*x^4 - 39056*a^4*b^3*x^3 - 44928*a^5*b^2*x^2 - 24320*a^6*b*x - 5120*a^7)*sqrt(b*x + a))/(a^3*x^7)]

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Sympy [A]  time = 28.2355, size = 236, normalized size = 1.45 \begin{align*} - \frac{a^{5}}{7 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{23 a^{4} \sqrt{b}}{28 x^{\frac{13}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{541 a^{3} b^{\frac{3}{2}}}{280 x^{\frac{11}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5249 a^{2} b^{\frac{5}{2}}}{2240 x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{6653 a b^{\frac{7}{2}}}{4480 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{1027 b^{\frac{9}{2}}}{2560 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{11}{2}}}{1024 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{9 b^{\frac{13}{2}}}{1024 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{9 b^{7} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{1024 a^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(9/2)/x**8,x)

[Out]

-a**5/(7*sqrt(b)*x**(15/2)*sqrt(a/(b*x) + 1)) - 23*a**4*sqrt(b)/(28*x**(13/2)*sqrt(a/(b*x) + 1)) - 541*a**3*b*
*(3/2)/(280*x**(11/2)*sqrt(a/(b*x) + 1)) - 5249*a**2*b**(5/2)/(2240*x**(9/2)*sqrt(a/(b*x) + 1)) - 6653*a*b**(7
/2)/(4480*x**(7/2)*sqrt(a/(b*x) + 1)) - 1027*b**(9/2)/(2560*x**(5/2)*sqrt(a/(b*x) + 1)) + 3*b**(11/2)/(1024*a*
x**(3/2)*sqrt(a/(b*x) + 1)) + 9*b**(13/2)/(1024*a**2*sqrt(x)*sqrt(a/(b*x) + 1)) - 9*b**7*asinh(sqrt(a)/(sqrt(b
)*sqrt(x)))/(1024*a**(5/2))

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Giac [A]  time = 1.22627, size = 194, normalized size = 1.19 \begin{align*} \frac{\frac{315 \, b^{8} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{315 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{8} - 2100 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{8} - 8393 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{8} + 9216 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{8} - 5943 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{8} + 2100 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{8} - 315 \, \sqrt{b x + a} a^{6} b^{8}}{a^{2} b^{7} x^{7}}}{35840 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^8,x, algorithm="giac")

[Out]

1/35840*(315*b^8*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (315*(b*x + a)^(13/2)*b^8 - 2100*(b*x + a)^(1
1/2)*a*b^8 - 8393*(b*x + a)^(9/2)*a^2*b^8 + 9216*(b*x + a)^(7/2)*a^3*b^8 - 5943*(b*x + a)^(5/2)*a^4*b^8 + 2100
*(b*x + a)^(3/2)*a^5*b^8 - 315*sqrt(b*x + a)*a^6*b^8)/(a^2*b^7*x^7))/b