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

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 163, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 = 0.01, size = 35, normalized size = 0.21 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.26, size = 119, normalized size = 0.73 \begin {gather*} -\frac {9 b^7 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}}-\frac {\sqrt {a+b x} \left (315 a^6-2100 a^5 (a+b x)+5943 a^4 (a+b x)^2-9216 a^3 (a+b x)^3+8393 a^2 (a+b x)^4+2100 a (a+b x)^5-315 (a+b x)^6\right )}{35840 a^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(9/2)/x^8,x]

[Out]

-1/35840*(Sqrt[a + b*x]*(315*a^6 - 2100*a^5*(a + b*x) + 5943*a^4*(a + b*x)^2 - 9216*a^3*(a + b*x)^3 + 8393*a^2
*(a + b*x)^4 + 2100*a*(a + b*x)^5 - 315*(a + b*x)^6))/(a^2*x^7) - (9*b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024
*a^(5/2))

________________________________________________________________________________________

fricas [A]  time = 1.42, size = 233, normalized size = 1.43 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

giac [A]  time = 0.95, size = 144, normalized size = 0.88 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 111, normalized size = 0.68 \begin {gather*} 2 \left (-\frac {9 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}+\frac {-\frac {9 \sqrt {b x +a}\, a^{4}}{2048}+\frac {15 \left (b x +a \right )^{\frac {3}{2}} a^{3}}{512}-\frac {849 \left (b x +a \right )^{\frac {5}{2}} a^{2}}{10240}+\frac {9 \left (b x +a \right )^{\frac {7}{2}} a}{70}-\frac {15 \left (b x +a \right )^{\frac {11}{2}}}{512 a}+\frac {9 \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{2}}-\frac {1199 \left (b x +a \right )^{\frac {9}{2}}}{10240}}{b^{7} x^{7}}\right ) b^{7} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 3.08, size = 229, normalized size = 1.40 \begin {gather*} \frac {9 \, b^{7} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2048 \, a^{\frac {5}{2}}} + \frac {315 \, {\left (b x + a\right )}^{\frac {13}{2}} b^{7} - 2100 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{7} - 8393 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{7} + 9216 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{7} - 5943 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{7} + 2100 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{7} - 315 \, \sqrt {b x + a} a^{6} b^{7}}{35840 \, {\left ({\left (b x + a\right )}^{7} a^{2} - 7 \, {\left (b x + a\right )}^{6} a^{3} + 21 \, {\left (b x + a\right )}^{5} a^{4} - 35 \, {\left (b x + a\right )}^{4} a^{5} + 35 \, {\left (b x + a\right )}^{3} a^{6} - 21 \, {\left (b x + a\right )}^{2} a^{7} + 7 \, {\left (b x + a\right )} a^{8} - a^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/2048*b^7*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/a^(5/2) + 1/35840*(315*(b*x + a)^(13/2)*b^
7 - 2100*(b*x + a)^(11/2)*a*b^7 - 8393*(b*x + a)^(9/2)*a^2*b^7 + 9216*(b*x + a)^(7/2)*a^3*b^7 - 5943*(b*x + a)
^(5/2)*a^4*b^7 + 2100*(b*x + a)^(3/2)*a^5*b^7 - 315*sqrt(b*x + a)*a^6*b^7)/((b*x + a)^7*a^2 - 7*(b*x + a)^6*a^
3 + 21*(b*x + a)^5*a^4 - 35*(b*x + a)^4*a^5 + 35*(b*x + a)^3*a^6 - 21*(b*x + a)^2*a^7 + 7*(b*x + a)*a^8 - a^9)

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 124, normalized size = 0.76 \begin {gather*} \frac {15\,a^3\,{\left (a+b\,x\right )}^{3/2}}{256\,x^7}-\frac {9\,a^4\,\sqrt {a+b\,x}}{1024\,x^7}-\frac {1199\,{\left (a+b\,x\right )}^{9/2}}{5120\,x^7}-\frac {849\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5120\,x^7}-\frac {15\,{\left (a+b\,x\right )}^{11/2}}{256\,a\,x^7}+\frac {9\,{\left (a+b\,x\right )}^{13/2}}{1024\,a^2\,x^7}+\frac {9\,a\,{\left (a+b\,x\right )}^{7/2}}{35\,x^7}+\frac {b^7\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,9{}\mathrm {i}}{1024\,a^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(9/2)/x^8,x)

[Out]

(15*a^3*(a + b*x)^(3/2))/(256*x^7) - (9*a^4*(a + b*x)^(1/2))/(1024*x^7) - (1199*(a + b*x)^(9/2))/(5120*x^7) -
(849*a^2*(a + b*x)^(5/2))/(5120*x^7) - (15*(a + b*x)^(11/2))/(256*a*x^7) + (9*(a + b*x)^(13/2))/(1024*a^2*x^7)
+ (b^7*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*9i)/(1024*a^(5/2)) + (9*a*(a + b*x)^(7/2))/(35*x^7)

________________________________________________________________________________________

sympy [A]  time = 22.20, size = 236, normalized size = 1.45 \begin {gather*} - \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 {gather*}

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))

________________________________________________________________________________________