3.5.60 \(\int \frac {1}{x^{3/2} (a+b x)^2} \, dx\) [460]

Optimal. Leaf size=56 \[ -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {44, 53, 65, 211} \begin {gather*} -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x)^2),x]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a + b*x)) - (3*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(5/2)

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 211

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

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.96 \begin {gather*} \frac {-2 a-3 b x}{a^2 \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x)^2),x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 8.48, size = 358, normalized size = 6.39 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {5}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{5 b^2 x^{\frac {5}{2}}},a\text {==}0\right \},\left \{\frac {-2}{a^2 \sqrt {x}},b\text {==}0\right \}\right \},\frac {-4 a \sqrt {-\frac {a}{b}}}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}-\frac {3 a \sqrt {x} \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}+\frac {3 a \sqrt {x} \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}-\frac {6 b x \sqrt {-\frac {a}{b}}}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}-\frac {3 b x^{\frac {3}{2}} \text {Log}\left [\sqrt {x}-\sqrt {-\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}+\frac {3 b x^{\frac {3}{2}} \text {Log}\left [\sqrt {x}+\sqrt {-\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {-\frac {a}{b}}+2 a^2 b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^(3/2)*(a + b*x)^2),x]')

[Out]

Piecewise[{{DirectedInfinity[1 / x ^ (5 / 2)], a == 0 && b == 0}, {-2 / (5 b ^ 2 x ^ (5 / 2)), a == 0}, {-2 /
(a ^ 2 Sqrt[x]), b == 0}}, -4 a Sqrt[-a / b] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[-a /
b]) - 3 a Sqrt[x] Log[Sqrt[x] - Sqrt[-a / b]] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[-a
/ b]) + 3 a Sqrt[x] Log[Sqrt[x] + Sqrt[-a / b]] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[
-a / b]) - 6 b x Sqrt[-a / b] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[-a / b]) - 3 b x ^
(3 / 2) Log[Sqrt[x] - Sqrt[-a / b]] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[-a / b]) + 3
b x ^ (3 / 2) Log[Sqrt[x] + Sqrt[-a / b]] / (2 a ^ 3 Sqrt[x] Sqrt[-a / b] + 2 a ^ 2 b x ^ (3 / 2) Sqrt[-a / b]
)]

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Maple [A]
time = 0.11, size = 47, normalized size = 0.84

method	result	size
derivativedivides	\(-\frac {2 b \left (\frac {\sqrt {x}}{2 b x +2 a}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(47\)
default	\(-\frac {2 b \left (\frac {\sqrt {x}}{2 b x +2 a}+\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(47\)
risch	\(-\frac {2}{a^{2} \sqrt {x}}-\frac {b \sqrt {x}}{a^{2} \left (b x +a \right )}-\frac {3 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 51, normalized size = 0.91 \begin {gather*} -\frac {3 \, b x + 2 \, a}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} - \frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.32, size = 147, normalized size = 2.62 \begin {gather*} \left [\frac {3 \, {\left (b x^{2} + a x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (3 \, b x + 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac {3 \, {\left (b x^{2} + a x\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (3 \, b x + 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} + a^{3} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 7.38, size = 384, normalized size = 6.86 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {3 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {4 a \sqrt {- \frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 b x \sqrt {- \frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 0.00, size = 71, normalized size = 1.27 \begin {gather*} 2 \left (\frac {-3 x b-2 a}{2 a^{2} \left (\sqrt {x} x b+\sqrt {x} a\right )}-\frac {3 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 a^{2} \sqrt {a b}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x+a)^2,x)

[Out]

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

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Mupad [B]
time = 0.12, size = 48, normalized size = 0.86 \begin {gather*} -\frac {\frac {2}{a}+\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a + b*x)^2),x)

[Out]

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

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