3.5.77 \(\int \frac {x^{3/2}}{(-a+b x)^2} \, dx\) [477]

Optimal. Leaf size=57 \[ \frac {3 \sqrt {x}}{b^2}+\frac {x^{3/2}}{b (a-b x)}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \]

[Out]

x^(3/2)/b/(-b*x+a)-3*arctanh(b^(1/2)*x^(1/2)/a^(1/2))*a^(1/2)/b^(5/2)+3*x^(1/2)/b^2

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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 52, 65, 214} \begin {gather*} -\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x^{3/2}}{b (a-b x)}+\frac {3 \sqrt {x}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(-a + b*x)^2,x]

[Out]

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

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 52

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 65

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 214

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x} (-3 a+2 b x)}{b^2 (-a+b x)}-\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(-a + b*x)^2,x]

[Out]

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

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Maple [A]
time = 0.13, size = 48, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (-b x +a \right )}+\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\)	\(48\)
default	\(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{2 \left (-b x +a \right )}+\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{2}}\)	\(48\)
risch	\(\frac {2 \sqrt {x}}{b^{2}}+\frac {a \left (-\frac {\sqrt {x}}{b x -a}-\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{b^{2}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(b*x-a)^2,x,method=_RETURNVERBOSE)

[Out]

2*x^(1/2)/b^2-2*a/b^2*(-1/2*x^(1/2)/(-b*x+a)+3/2/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2)))

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Maxima [A]
time = 0.50, size = 68, normalized size = 1.19 \begin {gather*} -\frac {a \sqrt {x}}{b^{3} x - a b^{2}} + \frac {3 \, a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x-a)^2,x, algorithm="maxima")

[Out]

-a*sqrt(x)/(b^3*x - a*b^2) + 3/2*a*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*b^2) + 2*sq
rt(x)/b^2

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Fricas [A]
time = 0.44, size = 138, normalized size = 2.42 \begin {gather*} \left [\frac {3 \, {\left (b x - a\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 x - 3 \, a\right )} \sqrt {x}}{2 \, {\left (b^{3} x - a b^{2}\right )}}, \frac {3 \, {\left (b x - a\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (2 \, b x - 3 \, a\right )} \sqrt {x}}{b^{3} x - a b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x-a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (49) = 98\).
time = 3.76, size = 301, normalized size = 5.28 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{2}} & \text {for}\: b = 0 \\- \frac {3 a^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {3 a^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} - \frac {6 a b \sqrt {x} \sqrt {\frac {a}{b}}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {3 a b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} - \frac {3 a b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} + \frac {4 b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{- 2 a b^{3} \sqrt {\frac {a}{b}} + 2 b^{4} x \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(b*x-a)**2,x)

[Out]

Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0)), (2*sqrt(x)/b**2, Eq(a, 0)), (2*x**(5/2)/(5*a**2), Eq(b, 0)), (-3
*a**2*log(sqrt(x) - sqrt(a/b))/(-2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt(a/b)) + 3*a**2*log(sqrt(x) + sqrt(a/b))/(-
2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt(a/b)) - 6*a*b*sqrt(x)*sqrt(a/b)/(-2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt(a/b))
+ 3*a*b*x*log(sqrt(x) - sqrt(a/b))/(-2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt(a/b)) - 3*a*b*x*log(sqrt(x) + sqrt(a/b
))/(-2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt(a/b)) + 4*b**2*x**(3/2)*sqrt(a/b)/(-2*a*b**3*sqrt(a/b) + 2*b**4*x*sqrt
(a/b)), True))

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Giac [A]
time = 0.59, size = 51, normalized size = 0.89 \begin {gather*} \frac {3 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b^{2}} - \frac {a \sqrt {x}}{{\left (b x - a\right )} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(b*x-a)^2,x, algorithm="giac")

[Out]

3*a*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b^2) - a*sqrt(x)/((b*x - a)*b^2) + 2*sqrt(x)/b^2

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Mupad [B]
time = 0.11, size = 47, normalized size = 0.82 \begin {gather*} \frac {2\,\sqrt {x}}{b^2}+\frac {a\,\sqrt {x}}{a\,b^2-b^3\,x}-\frac {3\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{b^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a - b*x)^2,x)

[Out]

(2*x^(1/2))/b^2 + (a*x^(1/2))/(a*b^2 - b^3*x) - (3*a^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/b^(5/2)

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