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

Optimal. Leaf size=70 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{5 a \sqrt{x}}{b^3}-\frac{x^{5/2}}{b (a+b x)}+\frac{5 x^{3/2}}{3 b^2} \]

[Out]

(-5*a*Sqrt[x])/b^3 + (5*x^(3/2))/(3*b^2) - x^(5/2)/(b*(a + b*x)) + (5*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]
])/b^(7/2)

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Rubi [A]  time = 0.0221569, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 50, 63, 205} \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}-\frac{5 a \sqrt{x}}{b^3}-\frac{x^{5/2}}{b (a+b x)}+\frac{5 x^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/(a + b*x)^2,x]

[Out]

(-5*a*Sqrt[x])/b^3 + (5*x^(3/2))/(3*b^2) - x^(5/2)/(b*(a + b*x)) + (5*a^(3/2)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]
])/b^(7/2)

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 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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{(a+b x)^2} \, dx &=-\frac{x^{5/2}}{b (a+b x)}+\frac{5 \int \frac{x^{3/2}}{a+b x} \, dx}{2 b}\\ &=\frac{5 x^{3/2}}{3 b^2}-\frac{x^{5/2}}{b (a+b x)}-\frac{(5 a) \int \frac{\sqrt{x}}{a+b x} \, dx}{2 b^2}\\ &=-\frac{5 a \sqrt{x}}{b^3}+\frac{5 x^{3/2}}{3 b^2}-\frac{x^{5/2}}{b (a+b x)}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 b^3}\\ &=-\frac{5 a \sqrt{x}}{b^3}+\frac{5 x^{3/2}}{3 b^2}-\frac{x^{5/2}}{b (a+b x)}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=-\frac{5 a \sqrt{x}}{b^3}+\frac{5 x^{3/2}}{3 b^2}-\frac{x^{5/2}}{b (a+b x)}+\frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0043399, size = 27, normalized size = 0.39 \[ \frac{2 x^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{b x}{a}\right )}{7 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a + b*x)^2,x]

[Out]

(2*x^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*x)/a)])/(7*a^2)

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Maple [A]  time = 0.01, size = 61, normalized size = 0.9 \begin{align*}{\frac{2}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-4\,{\frac{a\sqrt{x}}{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)/b^2-4*a*x^(1/2)/b^3-1/b^3*a^2*x^(1/2)/(b*x+a)+5/b^3*a^2/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(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(x^(5/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.63717, size = 366, normalized size = 5.23 \begin{align*} \left [\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, \frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (2 \, b^{2} x^{2} - 10 \, a b x - 15 \, a^{2}\right )} \sqrt{x}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*(a*b*x + a^2)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) + 2*(2*b^2*x^2 - 10*a*b*x
- 15*a^2)*sqrt(x))/(b^4*x + a*b^3), 1/3*(15*(a*b*x + a^2)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (2*b^2*x^2
- 10*a*b*x - 15*a^2)*sqrt(x))/(b^4*x + a*b^3)]

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Sympy [A]  time = 74.5236, size = 479, normalized size = 6.84 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{3}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b^{2}} & \text{for}\: a = 0 \\\frac{2 x^{\frac{7}{2}}}{7 a^{2}} & \text{for}\: b = 0 \\- \frac{30 i a^{\frac{5}{2}} b \sqrt{x} \sqrt{\frac{1}{b}}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} - \frac{20 i a^{\frac{3}{2}} b^{2} x^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} + \frac{4 i \sqrt{a} b^{3} x^{\frac{5}{2}} \sqrt{\frac{1}{b}}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} + \frac{15 a^{3} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} - \frac{15 a^{3} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} + \frac{15 a^{2} b x \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} - \frac{15 a^{2} b x \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{6 i a^{\frac{3}{2}} b^{4} \sqrt{\frac{1}{b}} + 6 i \sqrt{a} b^{5} x \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*b**2), Eq(a, 0)), (2*x**(7/2)/(7*a**2), Eq(b, 0)
), (-30*I*a**(5/2)*b*sqrt(x)*sqrt(1/b)/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) - 20*I*a**
(3/2)*b**2*x**(3/2)*sqrt(1/b)/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) + 4*I*sqrt(a)*b**3*
x**(5/2)*sqrt(1/b)/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) + 15*a**3*log(-I*sqrt(a)*sqrt(
1/b) + sqrt(x))/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) - 15*a**3*log(I*sqrt(a)*sqrt(1/b)
+ sqrt(x))/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) + 15*a**2*b*x*log(-I*sqrt(a)*sqrt(1/b
) + sqrt(x))/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)) - 15*a**2*b*x*log(I*sqrt(a)*sqrt(1/b
) + sqrt(x))/(6*I*a**(3/2)*b**4*sqrt(1/b) + 6*I*sqrt(a)*b**5*x*sqrt(1/b)), True))

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Giac [A]  time = 1.21441, size = 88, normalized size = 1.26 \begin{align*} \frac{5 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{a^{2} \sqrt{x}}{{\left (b x + a\right )} b^{3}} + \frac{2 \,{\left (b^{4} x^{\frac{3}{2}} - 6 \, a b^{3} \sqrt{x}\right )}}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a^2*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^3) - a^2*sqrt(x)/((b*x + a)*b^3) + 2/3*(b^4*x^(3/2) - 6*a*b^3*s
qrt(x))/b^6