3.4.59 \(\int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx\) [359]

Optimal. Leaf size=88 \[ -\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 211} \begin {gather*} -\frac {(3 A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(3/2)*(a + b*x)^2),x]

[Out]

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

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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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

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Mathematica [A]
time = 0.09, size = 67, normalized size = 0.76 \begin {gather*} \frac {-2 a A-3 A b x+a B x}{a^2 \sqrt {x} (a+b x)}+\frac {(-3 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(3/2)*(a + b*x)^2),x]

[Out]

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

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Maple [A]
time = 0.08, size = 64, normalized size = 0.73

method result size
derivativedivides \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) \(64\)
default \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) \(64\)
risch \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\sqrt {x}\, A b}{a^{2} \left (b x +a \right )}+\frac {\sqrt {x}\, B}{a \left (b x +a \right )}-\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A b}{a^{2} \sqrt {a b}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{a \sqrt {a b}}\) \(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 65, normalized size = 0.74 \begin {gather*} -\frac {2 \, A a - {\left (B a - 3 \, A b\right )} x}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} + \frac {{\left (B a - 3 \, A b\right )} \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((B*x+A)/x^(3/2)/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.40, size = 215, normalized size = 2.44 \begin {gather*} \left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (73) = 146\).
time = 8.10, size = 794, normalized size = 9.02 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {4 A a b \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 A b^{2} x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {2 B a b x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} 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((B*x+A)/x**(3/2)/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 1.03, size = 60, normalized size = 0.68 \begin {gather*} \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.43, size = 65, normalized size = 0.74 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {x\,\left (3\,A\,b-B\,a\right )}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b-B\,a\right )}{a^{5/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(3/2)*(a + b*x)^2),x)

[Out]

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

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