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

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

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Rubi [A]  time = 0.04, antiderivative size = 116, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [C]  time = 0.01, size = 35, normalized size = 0.30 \begin {gather*} -\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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.17, size = 92, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a+b x} \left (315 a^4-1155 a^3 (a+b x)+1533 a^2 (a+b x)^2-837 a (a+b x)^3+128 (a+b x)^4\right )}{64 x^4}-\frac {315}{64} \sqrt {a} b^4 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(9/2)/x^5,x]

[Out]

(Sqrt[a + b*x]*(315*a^4 - 1155*a^3*(a + b*x) + 1533*a^2*(a + b*x)^2 - 837*a*(a + b*x)^3 + 128*(a + b*x)^4))/(6
4*x^4) - (315*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64

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fricas [A]  time = 0.98, size = 177, normalized size = 1.53 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.22, size = 110, normalized size = 0.95 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.01, size = 85, normalized size = 0.73 \begin {gather*} 2 \left (\left (-\frac {315 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}+\frac {\frac {187 \sqrt {b x +a}\, a^{3}}{128}-\frac {643 \left (b x +a \right )^{\frac {3}{2}} a^{2}}{128}+\frac {765 \left (b x +a \right )^{\frac {5}{2}} a}{128}-\frac {325 \left (b x +a \right )^{\frac {7}{2}}}{128}}{b^{4} x^{4}}\right ) a +\sqrt {b x +a}\right ) b^{4} \end {gather*}

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*(b*x+a)^(5/2)*a-643/128*(b*x+a)^(3/2)*a^2+187/128*(b*x
+a)^(1/2)*a^3)/x^4/b^4-315/128*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)))

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maxima [A]  time = 2.96, size = 155, normalized size = 1.34 \begin {gather*} \frac {315}{128} \, \sqrt {a} b^{4} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x + a} b^{4} - \frac {325 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{4} - 765 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{4} + 643 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{4} - 187 \, \sqrt {b x + a} a^{4} b^{4}}{64 \, {\left ({\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} a + 6 \, {\left (b x + a\right )}^{2} a^{2} - 4 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

315/128*sqrt(a)*b^4*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) + 2*sqrt(b*x + a)*b^4 - 1/64*(325
*(b*x + a)^(7/2)*a*b^4 - 765*(b*x + a)^(5/2)*a^2*b^4 + 643*(b*x + a)^(3/2)*a^3*b^4 - 187*sqrt(b*x + a)*a^4*b^4
)/((b*x + a)^4 - 4*(b*x + a)^3*a + 6*(b*x + a)^2*a^2 - 4*(b*x + a)*a^3 + a^4)

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mupad [B]  time = 0.06, size = 94, normalized size = 0.81 \begin {gather*} 2\,b^4\,\sqrt {a+b\,x}+\frac {187\,a^4\,\sqrt {a+b\,x}}{64\,x^4}-\frac {643\,a^3\,{\left (a+b\,x\right )}^{3/2}}{64\,x^4}+\frac {765\,a^2\,{\left (a+b\,x\right )}^{5/2}}{64\,x^4}-\frac {325\,a\,{\left (a+b\,x\right )}^{7/2}}{64\,x^4}+\frac {\sqrt {a}\,b^4\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,315{}\mathrm {i}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(9/2)/x^5,x)

[Out]

2*b^4*(a + b*x)^(1/2) + (187*a^4*(a + b*x)^(1/2))/(64*x^4) - (643*a^3*(a + b*x)^(3/2))/(64*x^4) + (765*a^2*(a
+ b*x)^(5/2))/(64*x^4) + (a^(1/2)*b^4*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*315i)/64 - (325*a*(a + b*x)^(7/2))/(6
4*x^4)

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sympy [A]  time = 8.55, size = 182, normalized size = 1.57 \begin {gather*} - \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 {gather*}

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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