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

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

[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))

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Rubi [A]  time = 0.0510884, antiderivative size = 141, normalized size of antiderivative = 1., 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} \[ \frac{21 b^6 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{3/2}}-\frac{21 b^4 \sqrt{a+b x}}{256 x^2}-\frac{7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac{21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac{21 b^5 \sqrt{a+b x}}{512 a x}-\frac{3 b (a+b x)^{7/2}}{20 x^5}-\frac{(a+b x)^{9/2}}{6 x^6} \]

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 = 0.0248076, size = 35, normalized size = 0.25 \[ -\frac{2 b^6 (a+b x)^{11/2} \, _2F_1\left (\frac{11}{2},7;\frac{13}{2};\frac{b x}{a}+1\right )}{11 a^7} \]

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)

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Maple [A]  time = 0.013, size = 99, normalized size = 0.7 \begin{align*} 2\,{b}^{6} \left ({\frac{1}{{b}^{6}{x}^{6}} \left ( -{\frac{21\, \left ( bx+a \right ) ^{11/2}}{1024\,a}}-{\frac{667\, \left ( bx+a \right ) ^{9/2}}{3072}}+{\frac{843\,a \left ( bx+a \right ) ^{7/2}}{2560}}-{\frac{693\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{2560}}+{\frac{119\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{1024}}-{\frac{21\,{a}^{4}\sqrt{bx+a}}{1024}} \right ) }+{\frac{21}{1024\,{a}^{3/2}}{\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^7,x)

[Out]

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

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62402, size = 541, normalized size = 3.84 \begin{align*} \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{align*}

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)]

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Sympy [A]  time = 20.8717, size = 209, normalized size = 1.48 \begin{align*} - \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{align*}

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))

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Giac [A]  time = 1.246, size = 174, normalized size = 1.23 \begin{align*} -\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{align*}

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