3.321 \(\int \frac{(a+b x)^{9/2}}{x^5} \, dx\)

Optimal. Leaf size=116 \[ -\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{105 b^3 (a+b x)^{3/2}}{64 x}+\frac{315}{64} b^4 \sqrt{a+b x}-\frac{315}{64} \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{9/2}}{4 x^4}-\frac{3 b (a+b x)^{7/2}}{8 x^3} \]

[Out]

(315*b^4*Sqrt[a + b*x])/64 - (105*b^3*(a + b*x)^(3/2))/(64*x) - (21*b^2*(a + b*x)^(5/2))/(32*x^2) - (3*b*(a +
b*x)^(7/2))/(8*x^3) - (a + b*x)^(9/2)/(4*x^4) - (315*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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Rubi [A]  time = 0.0366599, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 208} \[ -\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{105 b^3 (a+b x)^{3/2}}{64 x}+\frac{315}{64} b^4 \sqrt{a+b x}-\frac{315}{64} \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{9/2}}{4 x^4}-\frac{3 b (a+b x)^{7/2}}{8 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(9/2)/x^5,x]

[Out]

(315*b^4*Sqrt[a + b*x])/64 - (105*b^3*(a + b*x)^(3/2))/(64*x) - (21*b^2*(a + b*x)^(5/2))/(32*x^2) - (3*b*(a +
b*x)^(7/2))/(8*x^3) - (a + b*x)^(9/2)/(4*x^4) - (315*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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^5} \, dx &=-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{8} (9 b) \int \frac{(a+b x)^{7/2}}{x^4} \, dx\\ &=-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{16} \left (21 b^2\right ) \int \frac{(a+b x)^{5/2}}{x^3} \, dx\\ &=-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{64} \left (105 b^3\right ) \int \frac{(a+b x)^{3/2}}{x^2} \, dx\\ &=-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{128} \left (315 b^4\right ) \int \frac{\sqrt{a+b x}}{x} \, dx\\ &=\frac{315}{64} b^4 \sqrt{a+b x}-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{128} \left (315 a b^4\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx\\ &=\frac{315}{64} b^4 \sqrt{a+b x}-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}+\frac{1}{64} \left (315 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )\\ &=\frac{315}{64} b^4 \sqrt{a+b x}-\frac{105 b^3 (a+b x)^{3/2}}{64 x}-\frac{21 b^2 (a+b x)^{5/2}}{32 x^2}-\frac{3 b (a+b x)^{7/2}}{8 x^3}-\frac{(a+b x)^{9/2}}{4 x^4}-\frac{315}{64} \sqrt{a} b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0468802, size = 35, normalized size = 0.3 \[ -\frac{2 b^4 (a+b x)^{11/2} \, _2F_1\left (5,\frac{11}{2};\frac{13}{2};\frac{b x}{a}+1\right )}{11 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2)/x^5,x]

[Out]

(-2*b^4*(a + b*x)^(11/2)*Hypergeometric2F1[5, 11/2, 13/2, 1 + (b*x)/a])/(11*a^5)

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Maple [A]  time = 0.012, size = 85, normalized size = 0.7 \begin{align*} 2\,{b}^{4} \left ( \sqrt{bx+a}+a \left ({\frac{1}{{b}^{4}{x}^{4}} \left ( -{\frac{325\, \left ( bx+a \right ) ^{7/2}}{128}}+{\frac{765\,a \left ( bx+a \right ) ^{5/2}}{128}}-{\frac{643\,{a}^{2} \left ( bx+a \right ) ^{3/2}}{128}}+{\frac{187\,\sqrt{bx+a}{a}^{3}}{128}} \right ) }-{\frac{315}{128\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(9/2)/x^5,x)

[Out]

2*b^4*((b*x+a)^(1/2)+a*((-325/128*(b*x+a)^(7/2)+765/128*a*(b*x+a)^(5/2)-643/128*a^2*(b*x+a)^(3/2)+187/128*(b*x
+a)^(1/2)*a^3)/b^4/x^4-315/128*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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53474, size = 435, normalized size = 3.75 \begin{align*} \left [\frac{315 \, \sqrt{a} b^{4} x^{4} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x + a}}{128 \, x^{4}}, \frac{315 \, \sqrt{-a} b^{4} x^{4} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x + a}}{64 \, x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^5,x, algorithm="fricas")

[Out]

[1/128*(315*sqrt(a)*b^4*x^4*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(128*b^4*x^4 - 325*a*b^3*x^3 - 21
0*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4)*sqrt(b*x + a))/x^4, 1/64*(315*sqrt(-a)*b^4*x^4*arctan(sqrt(b*x + a)*sqrt(
-a)/a) + (128*b^4*x^4 - 325*a*b^3*x^3 - 210*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4)*sqrt(b*x + a))/x^4]

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Sympy [A]  time = 12.2001, size = 182, normalized size = 1.57 \begin{align*} - \frac{315 \sqrt{a} b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{64} - \frac{a^{5}}{4 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{13 a^{4} \sqrt{b}}{8 x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{149 a^{3} b^{\frac{3}{2}}}{32 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{535 a^{2} b^{\frac{5}{2}}}{64 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{197 a b^{\frac{7}{2}}}{64 \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{9}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(9/2)/x**5,x)

[Out]

-315*sqrt(a)*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/64 - a**5/(4*sqrt(b)*x**(9/2)*sqrt(a/(b*x) + 1)) - 13*a**4*
sqrt(b)/(8*x**(7/2)*sqrt(a/(b*x) + 1)) - 149*a**3*b**(3/2)/(32*x**(5/2)*sqrt(a/(b*x) + 1)) - 535*a**2*b**(5/2)
/(64*x**(3/2)*sqrt(a/(b*x) + 1)) - 197*a*b**(7/2)/(64*sqrt(x)*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*sqrt(x)/sqrt(a/(
b*x) + 1)

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Giac [A]  time = 1.22471, size = 149, normalized size = 1.28 \begin{align*} \frac{\frac{315 \, a b^{5} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 128 \, \sqrt{b x + a} b^{5} - \frac{325 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{5} - 765 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{5} + 643 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{5} - 187 \, \sqrt{b x + a} a^{4} b^{5}}{b^{4} x^{4}}}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^5,x, algorithm="giac")

[Out]

1/64*(315*a*b^5*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 128*sqrt(b*x + a)*b^5 - (325*(b*x + a)^(7/2)*a*b^5 -
765*(b*x + a)^(5/2)*a^2*b^5 + 643*(b*x + a)^(3/2)*a^3*b^5 - 187*sqrt(b*x + a)*a^4*b^5)/(b^4*x^4))/b