3.530 \(\int \frac{A+B x}{\sqrt{x} (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=60 \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0204526, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {78, 63, 217, 206} \[ \frac{2 \sqrt{x} (A b-a B)}{a b \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

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] || LtQ
[p, n]))))

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0581439, size = 76, normalized size = 1.27 \[ \frac{2 a^{3/2} B \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )+2 \sqrt{b} \sqrt{x} (A b-a B)}{a b^{3/2} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b]*(A*b - a*B)*Sqrt[x] + 2*a^(3/2)*B*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a*b^(3/2)*
Sqrt[a + b*x])

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Maple [B]  time = 0.013, size = 121, normalized size = 2. \begin{align*}{\frac{1}{a} \left ( B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ) xab+2\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+B\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a \right ){\frac{1}{\sqrt{b}}}} \right ){a}^{2}-2\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ) \sqrt{x}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^(3/2)/x^(1/2),x)

[Out]

(B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a*b+2*A*b^(3/2)*(x*(b*x+a))^(1/2)+B*ln(1/2*(2*(x*(b
*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^2-2*B*a*b^(1/2)*(x*(b*x+a))^(1/2))/a*x^(1/2)/(x*(b*x+a))^(1/2)/b^(3/2
)/(b*x+a)^(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((B*x+A)/(b*x+a)^(3/2)/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60359, size = 378, normalized size = 6.3 \begin{align*} \left [\frac{{\left (B a b x + B a^{2}\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} \sqrt{b x + a} \sqrt{x}}{a b^{3} x + a^{2} b^{2}}, -\frac{2 \,{\left ({\left (B a b x + B a^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (B a b - A b^{2}\right )} \sqrt{b x + a} \sqrt{x}\right )}}{a b^{3} x + a^{2} b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 15.7603, size = 68, normalized size = 1.13 \begin{align*} \frac{2 A}{a \sqrt{b} \sqrt{\frac{a}{b x} + 1}} + B \left (\frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{2 \sqrt{x}}{\sqrt{a} b \sqrt{1 + \frac{b x}{a}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A/(a*sqrt(b)*sqrt(a/(b*x) + 1)) + B*(2*asinh(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2) - 2*sqrt(x)/(sqrt(a)*b*sqrt(1
+ b*x/a)))

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Giac [B]  time = 87.5327, size = 131, normalized size = 2.18 \begin{align*} -\frac{B \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{\sqrt{b}{\left | b \right |}} - \frac{4 \,{\left (B a \sqrt{b} - A b^{\frac{3}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*log((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/(sqrt(b)*abs(b)) - 4*(B*a*sqrt(b) - A*b^(3/2))/(((
sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*abs(b))