3.540 \(\int \frac{\sqrt{x} (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=82 \[ \frac{2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}-\frac{2 B \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0261501, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 47, 63, 217, 206} \[ \frac{2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}-\frac{2 B \sqrt{x}}{b^2 \sqrt{a+b x}}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x]*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0866114, size = 95, normalized size = 1.16 \[ \frac{2 \sqrt{b} \sqrt{x} \left (-3 a^2 B-4 a b B x+A b^2 x\right )+6 a^{3/2} B (a+b x) \sqrt{\frac{b x}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{3 a b^{5/2} (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x]*(A + B*x))/(a + b*x)^(5/2),x]

[Out]

(2*Sqrt[b]*Sqrt[x]*(-3*a^2*B + A*b^2*x - 4*a*b*B*x) + 6*a^(3/2)*B*(a + b*x)*Sqrt[1 + (b*x)/a]*ArcSinh[(Sqrt[b]
*Sqrt[x])/Sqrt[a]])/(3*a*b^(5/2)*(a + b*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 182, normalized size = 2.2 \begin{align*}{\frac{1}{3\,a} \left ( 3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{2}+2\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x+6\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b-8\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}xa+3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}-6\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{2} \right ) \sqrt{x}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{b}^{-{\frac{5}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*x^(1/2)/(b*x+a)^(5/2),x)

[Out]

1/3*(3*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*a*b^2+2*A*(x*(b*x+a))^(1/2)*b^(5/2)*x+6*B*l
n(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x*a^2*b-8*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a+3*B*ln(1/2*(2*(
x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^3-6*B*(x*(b*x+a))^(1/2)*b^(1/2)*a^2)*x^(1/2)/a/(x*(b*x+a))^(1/2)/
b^(5/2)/(b*x+a)^(3/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.02089, size = 537, normalized size = 6.55 \begin{align*} \left [\frac{3 \,{\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (3 \, B a^{2} b +{\left (4 \, B a b^{2} - A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac{2 \,{\left (3 \,{\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (3 \, B a^{2} b +{\left (4 \, B a b^{2} - A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}\right )}}{3 \,{\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/3*(3*(B*a*b^2*x^2 + 2*B*a^2*b*x + B*a^3)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(3*B*
a^2*b + (4*B*a*b^2 - A*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(a*b^5*x^2 + 2*a^2*b^4*x + a^3*b^3), -2/3*(3*(B*a*b^2*x^
2 + 2*B*a^2*b*x + B*a^3)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (3*B*a^2*b + (4*B*a*b^2 - A*b^3
)*x)*sqrt(b*x + a)*sqrt(x))/(a*b^5*x^2 + 2*a^2*b^4*x + a^3*b^3)]

________________________________________________________________________________________

Sympy [B]  time = 24.4345, size = 376, normalized size = 4.59 \begin{align*} \frac{2 A x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}} + B \left (\frac{6 a^{\frac{39}{2}} b^{11} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{6 a^{\frac{37}{2}} b^{12} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{6 a^{19} b^{\frac{23}{2}} x^{14}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{8 a^{18} b^{\frac{25}{2}} x^{15}}{3 a^{\frac{39}{2}} b^{\frac{27}{2}} x^{\frac{27}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{37}{2}} b^{\frac{29}{2}} x^{\frac{29}{2}} \sqrt{1 + \frac{b x}{a}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x**(1/2)/(b*x+a)**(5/2),x)

[Out]

2*A*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*x*sqrt(1 + b*x/a)) + B*(6*a**(39/2)*b**11*x**(27/2)*sq
rt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b*
*(29/2)*x**(29/2)*sqrt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a)
)/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) - 6*a**1
9*b**(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 +
b*x/a)) - 8*a**18*b**(25/2)*x**15/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**
(29/2)*sqrt(1 + b*x/a)))

________________________________________________________________________________________

Giac [B]  time = 93.1678, size = 300, normalized size = 3.66 \begin{align*} -\frac{B{\left | b \right |} \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac{7}{2}}} - \frac{4 \,{\left (6 \, B a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 6 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 3 \, A{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 4 \, B a^{3} b^{\frac{5}{2}}{\left | b \right |} - A a^{2} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*x^(1/2)/(b*x+a)^(5/2),x, algorithm="giac")

[Out]

-B*abs(b)*log((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^(7/2) - 4/3*(6*B*a*(sqrt(b*x + a)*sqrt(b)
- sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(b) + 6*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(
3/2)*abs(b) - 3*A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 4*B*a^3*b^(5/2)*abs(b)
- A*a^2*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^3)