3.4.20 \(\int \frac {(a+b x)^{9/2}}{x^4} \, dx\)

Optimal. Leaf size=114 \[ -\frac {105}{8} a^{3/2} b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+\frac {35}{8} b^3 (a+b x)^{3/2}+\frac {105}{8} a b^3 \sqrt {a+b x}-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {(a+b x)^{9/2}}{3 x^3}-\frac {3 b (a+b x)^{7/2}}{4 x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 114, 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 {105}{8} a^{3/2} b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {21 b^2 (a+b x)^{5/2}}{8 x}+\frac {35}{8} b^3 (a+b x)^{3/2}+\frac {105}{8} a b^3 \sqrt {a+b x}-\frac {(a+b x)^{9/2}}{3 x^3}-\frac {3 b (a+b x)^{7/2}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(9/2)/x^4,x]

[Out]

(105*a*b^3*Sqrt[a + b*x])/8 + (35*b^3*(a + b*x)^(3/2))/8 - (21*b^2*(a + b*x)^(5/2))/(8*x) - (3*b*(a + b*x)^(7/
2))/(4*x^2) - (a + b*x)^(9/2)/(3*x^3) - (105*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/8

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^4} \, dx &=-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{2} (3 b) \int \frac {(a+b x)^{7/2}}{x^3} \, dx\\ &=-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{8} \left (21 b^2\right ) \int \frac {(a+b x)^{5/2}}{x^2} \, dx\\ &=-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{16} \left (105 b^3\right ) \int \frac {(a+b x)^{3/2}}{x} \, dx\\ &=\frac {35}{8} b^3 (a+b x)^{3/2}-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{16} \left (105 a b^3\right ) \int \frac {\sqrt {a+b x}}{x} \, dx\\ &=\frac {105}{8} a b^3 \sqrt {a+b x}+\frac {35}{8} b^3 (a+b x)^{3/2}-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{16} \left (105 a^2 b^3\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx\\ &=\frac {105}{8} a b^3 \sqrt {a+b x}+\frac {35}{8} b^3 (a+b x)^{3/2}-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}+\frac {1}{8} \left (105 a^2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )\\ &=\frac {105}{8} a b^3 \sqrt {a+b x}+\frac {35}{8} b^3 (a+b x)^{3/2}-\frac {21 b^2 (a+b x)^{5/2}}{8 x}-\frac {3 b (a+b x)^{7/2}}{4 x^2}-\frac {(a+b x)^{9/2}}{3 x^3}-\frac {105}{8} a^{3/2} b^3 \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 = 0.31 \begin {gather*} \frac {2 b^3 (a+b x)^{11/2} \, _2F_1\left (4,\frac {11}{2};\frac {13}{2};\frac {b x}{a}+1\right )}{11 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2)/x^4,x]

[Out]

(2*b^3*(a + b*x)^(11/2)*Hypergeometric2F1[4, 11/2, 13/2, 1 + (b*x)/a])/(11*a^4)

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IntegrateAlgebraic [A]  time = 0.14, size = 92, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a+b x} \left (-315 a^4+840 a^3 (a+b x)-693 a^2 (a+b x)^2+144 a (a+b x)^3+16 (a+b x)^4\right )}{24 x^3}-\frac {105}{8} a^{3/2} b^3 \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^4,x]

[Out]

(Sqrt[a + b*x]*(-315*a^4 + 840*a^3*(a + b*x) - 693*a^2*(a + b*x)^2 + 144*a*(a + b*x)^3 + 16*(a + b*x)^4))/(24*
x^3) - (105*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/8

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fricas [A]  time = 0.93, size = 178, normalized size = 1.56 \begin {gather*} \left [\frac {315 \, a^{\frac {3}{2}} b^{3} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt {b x + a}}{48 \, x^{3}}, \frac {315 \, \sqrt {-a} a b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt {b x + a}}{24 \, x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(315*a^(3/2)*b^3*x^3*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(16*b^4*x^4 + 208*a*b^3*x^3 - 165*
a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sqrt(b*x + a))/x^3, 1/24*(315*sqrt(-a)*a*b^3*x^3*arctan(sqrt(b*x + a)*sqrt(-
a)/a) + (16*b^4*x^4 + 208*a*b^3*x^3 - 165*a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sqrt(b*x + a))/x^3]

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giac [A]  time = 1.10, size = 112, normalized size = 0.98 \begin {gather*} \frac {\frac {315 \, a^{2} b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 16 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{4} + 192 \, \sqrt {b x + a} a b^{4} - \frac {165 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{4} - 280 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{4} + 123 \, \sqrt {b x + a} a^{4} b^{4}}{b^{3} x^{3}}}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^4,x, algorithm="giac")

[Out]

1/24*(315*a^2*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 16*(b*x + a)^(3/2)*b^4 + 192*sqrt(b*x + a)*a*b^4 -
(165*(b*x + a)^(5/2)*a^2*b^4 - 280*(b*x + a)^(3/2)*a^3*b^4 + 123*sqrt(b*x + a)*a^4*b^4)/(b^3*x^3))/b

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maple [A]  time = 0.01, size = 87, normalized size = 0.76 \begin {gather*} 2 \left (\left (-\frac {105 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {-\frac {41 \sqrt {b x +a}\, a^{2}}{16}+\frac {35 \left (b x +a \right )^{\frac {3}{2}} a}{6}-\frac {55 \left (b x +a \right )^{\frac {5}{2}}}{16}}{b^{3} x^{3}}\right ) a^{2}+4 \sqrt {b x +a}\, a +\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}\right ) b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(9/2)/x^4,x)

[Out]

2*b^3*(1/3*(b*x+a)^(3/2)+4*(b*x+a)^(1/2)*a+a^2*((-55/16*(b*x+a)^(5/2)+35/6*(b*x+a)^(3/2)*a-41/16*(b*x+a)^(1/2)
*a^2)/x^3/b^3-105/16*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)))

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maxima [A]  time = 2.94, size = 145, normalized size = 1.27 \begin {gather*} \frac {105}{16} \, a^{\frac {3}{2}} b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} + 8 \, \sqrt {b x + a} a b^{3} - \frac {165 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{3} - 280 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{3} + 123 \, \sqrt {b x + a} a^{4} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2} - a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2)/x^4,x, algorithm="maxima")

[Out]

105/16*a^(3/2)*b^3*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) + 2/3*(b*x + a)^(3/2)*b^3 + 8*sqrt
(b*x + a)*a*b^3 - 1/24*(165*(b*x + a)^(5/2)*a^2*b^3 - 280*(b*x + a)^(3/2)*a^3*b^3 + 123*sqrt(b*x + a)*a^4*b^3)
/((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2 - a^3)

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mupad [B]  time = 0.12, size = 131, normalized size = 1.15 \begin {gather*} \frac {2\,b^3\,{\left (a+b\,x\right )}^{3/2}}{3}+\frac {\frac {41\,a^4\,b^3\,\sqrt {a+b\,x}}{8}-\frac {35\,a^3\,b^3\,{\left (a+b\,x\right )}^{3/2}}{3}+\frac {55\,a^2\,b^3\,{\left (a+b\,x\right )}^{5/2}}{8}}{3\,a\,{\left (a+b\,x\right )}^2-3\,a^2\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^3+a^3}+8\,a\,b^3\,\sqrt {a+b\,x}+\frac {a^{3/2}\,b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,105{}\mathrm {i}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(9/2)/x^4,x)

[Out]

(2*b^3*(a + b*x)^(3/2))/3 + ((41*a^4*b^3*(a + b*x)^(1/2))/8 - (35*a^3*b^3*(a + b*x)^(3/2))/3 + (55*a^2*b^3*(a
+ b*x)^(5/2))/8)/(3*a*(a + b*x)^2 - 3*a^2*(a + b*x) - (a + b*x)^3 + a^3) + (a^(3/2)*b^3*atan(((a + b*x)^(1/2)*
1i)/a^(1/2))*105i)/8 + 8*a*b^3*(a + b*x)^(1/2)

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sympy [A]  time = 7.91, size = 184, normalized size = 1.61 \begin {gather*} - \frac {105 a^{\frac {3}{2}} b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8} - \frac {a^{5}}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {29 a^{4} \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {215 a^{3} b^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {43 a^{2} b^{\frac {5}{2}}}{24 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {28 a b^{\frac {7}{2}} \sqrt {x}}{3 \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {9}{2}} x^{\frac {3}{2}}}{3 \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**4,x)

[Out]

-105*a**(3/2)*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/8 - a**5/(3*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) - 29*a**4*
sqrt(b)/(12*x**(5/2)*sqrt(a/(b*x) + 1)) - 215*a**3*b**(3/2)/(24*x**(3/2)*sqrt(a/(b*x) + 1)) + 43*a**2*b**(5/2)
/(24*sqrt(x)*sqrt(a/(b*x) + 1)) + 28*a*b**(7/2)*sqrt(x)/(3*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*x**(3/2)/(3*sqrt(a/
(b*x) + 1))

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