3.4.23 \(\int \frac {(a+b x)^{9/2}}{x^7} \, dx\)

Optimal. Leaf size=141 \[ \frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {3 b (a+b x)^{7/2}}{20 x^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}}-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(9/2)/x^7,x]

[Out]

(-21*b^4*Sqrt[a + b*x])/(256*x^2) - (21*b^5*Sqrt[a + b*x])/(512*a*x) - (7*b^3*(a + b*x)^(3/2))/(64*x^3) - (21*
b^2*(a + b*x)^(5/2))/(160*x^4) - (3*b*(a + b*x)^(7/2))/(20*x^5) - (a + b*x)^(9/2)/(6*x^6) + (21*b^6*ArcTanh[Sq
rt[a + b*x]/Sqrt[a]])/(512*a^(3/2))

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^7} \, dx &=-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{4} (3 b) \int \frac {(a+b x)^{7/2}}{x^6} \, dx\\ &=-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{40} \left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^5} \, dx\\ &=-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{64} \left (21 b^3\right ) \int \frac {(a+b x)^{3/2}}{x^4} \, dx\\ &=-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{128} \left (21 b^4\right ) \int \frac {\sqrt {a+b x}}{x^3} \, dx\\ &=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {1}{512} \left (21 b^5\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx\\ &=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {\left (21 b^6\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{1024 a}\\ &=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}-\frac {\left (21 b^5\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{512 a}\\ &=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 35, normalized size = 0.25 \begin {gather*} -\frac {2 b^6 (a+b x)^{11/2} \, _2F_1\left (\frac {11}{2},7;\frac {13}{2};\frac {b x}{a}+1\right )}{11 a^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2)/x^7,x]

[Out]

(-2*b^6*(a + b*x)^(11/2)*Hypergeometric2F1[11/2, 7, 13/2, 1 + (b*x)/a])/(11*a^7)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.23, size = 107, normalized size = 0.76 \begin {gather*} \frac {21 b^6 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}}-\frac {\sqrt {a+b x} \left (315 a^5-1785 a^4 (a+b x)+4158 a^3 (a+b x)^2-5058 a^2 (a+b x)^3+3335 a (a+b x)^4+315 (a+b x)^5\right )}{7680 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(9/2)/x^7,x]

[Out]

-1/7680*(Sqrt[a + b*x]*(315*a^5 - 1785*a^4*(a + b*x) + 4158*a^3*(a + b*x)^2 - 5058*a^2*(a + b*x)^3 + 3335*a*(a
+ b*x)^4 + 315*(a + b*x)^5))/(a*x^6) + (21*b^6*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(3/2))

________________________________________________________________________________________

fricas [A]  time = 1.27, size = 211, normalized size = 1.50 \begin {gather*} \left [\frac {315 \, \sqrt {a} b^{6} x^{6} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{15360 \, a^{2} x^{6}}, -\frac {315 \, \sqrt {-a} b^{6} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{7680 \, a^{2} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^7,x, algorithm="fricas")

[Out]

[1/15360*(315*sqrt(a)*b^6*x^6*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*(315*a*b^5*x^5 + 4910*a^2*b^4*x
^4 + 11432*a^3*b^3*x^3 + 12144*a^4*b^2*x^2 + 6272*a^5*b*x + 1280*a^6)*sqrt(b*x + a))/(a^2*x^6), -1/7680*(315*s
qrt(-a)*b^6*x^6*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (315*a*b^5*x^5 + 4910*a^2*b^4*x^4 + 11432*a^3*b^3*x^3 + 121
44*a^4*b^2*x^2 + 6272*a^5*b*x + 1280*a^6)*sqrt(b*x + a))/(a^2*x^6)]

________________________________________________________________________________________

giac [A]  time = 1.04, size = 129, normalized size = 0.91 \begin {gather*} -\frac {\frac {315 \, b^{7} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{7} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{7} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{7} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{7} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{7} + 315 \, \sqrt {b x + a} a^{5} b^{7}}{a b^{6} x^{6}}}{7680 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^7,x, algorithm="giac")

[Out]

-1/7680*(315*b^7*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b*x + a)^(11/2)*b^7 + 3335*(b*x + a)^(9/2
)*a*b^7 - 5058*(b*x + a)^(7/2)*a^2*b^7 + 4158*(b*x + a)^(5/2)*a^3*b^7 - 1785*(b*x + a)^(3/2)*a^4*b^7 + 315*sqr
t(b*x + a)*a^5*b^7)/(a*b^6*x^6))/b

________________________________________________________________________________________

maple [A]  time = 0.01, size = 99, normalized size = 0.70 \begin {gather*} 2 \left (\frac {21 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {3}{2}}}+\frac {-\frac {21 \sqrt {b x +a}\, a^{4}}{1024}+\frac {119 \left (b x +a \right )^{\frac {3}{2}} a^{3}}{1024}-\frac {693 \left (b x +a \right )^{\frac {5}{2}} a^{2}}{2560}+\frac {843 \left (b x +a \right )^{\frac {7}{2}} a}{2560}-\frac {21 \left (b x +a \right )^{\frac {11}{2}}}{1024 a}-\frac {667 \left (b x +a \right )^{\frac {9}{2}}}{3072}}{b^{6} x^{6}}\right ) b^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(9/2)/x^7,x)

[Out]

2*b^6*((-21/1024/a*(b*x+a)^(11/2)-667/3072*(b*x+a)^(9/2)+843/2560*(b*x+a)^(7/2)*a-693/2560*(b*x+a)^(5/2)*a^2+1
19/1024*(b*x+a)^(3/2)*a^3-21/1024*(b*x+a)^(1/2)*a^4)/x^6/b^6+21/1024*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2))

________________________________________________________________________________________

maxima [A]  time = 3.03, size = 198, normalized size = 1.40 \begin {gather*} -\frac {21 \, b^{6} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{1024 \, a^{\frac {3}{2}}} - \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{6} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{6} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{6} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{6} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{6} + 315 \, \sqrt {b x + a} a^{5} b^{6}}{7680 \, {\left ({\left (b x + a\right )}^{6} a - 6 \, {\left (b x + a\right )}^{5} a^{2} + 15 \, {\left (b x + a\right )}^{4} a^{3} - 20 \, {\left (b x + a\right )}^{3} a^{4} + 15 \, {\left (b x + a\right )}^{2} a^{5} - 6 \, {\left (b x + a\right )} a^{6} + a^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^7,x, algorithm="maxima")

[Out]

-21/1024*b^6*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/a^(3/2) - 1/7680*(315*(b*x + a)^(11/2)*b
^6 + 3335*(b*x + a)^(9/2)*a*b^6 - 5058*(b*x + a)^(7/2)*a^2*b^6 + 4158*(b*x + a)^(5/2)*a^3*b^6 - 1785*(b*x + a)
^(3/2)*a^4*b^6 + 315*sqrt(b*x + a)*a^5*b^6)/((b*x + a)^6*a - 6*(b*x + a)^5*a^2 + 15*(b*x + a)^4*a^3 - 20*(b*x
+ a)^3*a^4 + 15*(b*x + a)^2*a^5 - 6*(b*x + a)*a^6 + a^7)

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 109, normalized size = 0.77 \begin {gather*} \frac {119\,a^3\,{\left (a+b\,x\right )}^{3/2}}{512\,x^6}-\frac {21\,a^4\,\sqrt {a+b\,x}}{512\,x^6}-\frac {667\,{\left (a+b\,x\right )}^{9/2}}{1536\,x^6}-\frac {693\,a^2\,{\left (a+b\,x\right )}^{5/2}}{1280\,x^6}-\frac {21\,{\left (a+b\,x\right )}^{11/2}}{512\,a\,x^6}+\frac {843\,a\,{\left (a+b\,x\right )}^{7/2}}{1280\,x^6}-\frac {b^6\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,21{}\mathrm {i}}{512\,a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(9/2)/x^7,x)

[Out]

(119*a^3*(a + b*x)^(3/2))/(512*x^6) - (21*a^4*(a + b*x)^(1/2))/(512*x^6) - (667*(a + b*x)^(9/2))/(1536*x^6) -
(693*a^2*(a + b*x)^(5/2))/(1280*x^6) - (21*(a + b*x)^(11/2))/(512*a*x^6) - (b^6*atan(((a + b*x)^(1/2)*1i)/a^(1
/2))*21i)/(512*a^(3/2)) + (843*a*(a + b*x)^(7/2))/(1280*x^6)

________________________________________________________________________________________

sympy [A]  time = 15.69, size = 209, normalized size = 1.48 \begin {gather*} - \frac {a^{5}}{6 \sqrt {b} x^{\frac {13}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {59 a^{4} \sqrt {b}}{60 x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1151 a^{3} b^{\frac {3}{2}}}{480 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {2947 a^{2} b^{\frac {5}{2}}}{960 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {8171 a b^{\frac {7}{2}}}{3840 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1045 b^{\frac {9}{2}}}{1536 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {21 b^{\frac {11}{2}}}{512 a \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {21 b^{6} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{512 a^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(9/2)/x**7,x)

[Out]

-a**5/(6*sqrt(b)*x**(13/2)*sqrt(a/(b*x) + 1)) - 59*a**4*sqrt(b)/(60*x**(11/2)*sqrt(a/(b*x) + 1)) - 1151*a**3*b
**(3/2)/(480*x**(9/2)*sqrt(a/(b*x) + 1)) - 2947*a**2*b**(5/2)/(960*x**(7/2)*sqrt(a/(b*x) + 1)) - 8171*a*b**(7/
2)/(3840*x**(5/2)*sqrt(a/(b*x) + 1)) - 1045*b**(9/2)/(1536*x**(3/2)*sqrt(a/(b*x) + 1)) - 21*b**(11/2)/(512*a*s
qrt(x)*sqrt(a/(b*x) + 1)) + 21*b**6*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(512*a**(3/2))

________________________________________________________________________________________