3.2238 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]

[Out]

-((b*B*d - 4*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b*
e) - ((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(
5/2)*e^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.106625, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {80, 50, 63, 217, 206} \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]

[Out]

-((b*B*d - 4*A*b*e + 3*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*b^2*e) + (B*Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b*
e) - ((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4*b^(
5/2)*e^(3/2))

Rule 80

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

Rule 50

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 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{d+e x}}{\sqrt{a+b x}} \, dx &=\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e}+\frac{\left (2 A b e-B \left (\frac{b d}{2}+\frac{3 a e}{2}\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x}} \, dx}{2 b e}\\ &=-\frac{(b B d-4 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e}-\frac{((b d-a e) (b B d-4 A b e+3 a B e)) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{8 b^2 e}\\ &=-\frac{(b B d-4 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e}-\frac{((b d-a e) (b B d-4 A b e+3 a B e)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{4 b^3 e}\\ &=-\frac{(b B d-4 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e}-\frac{((b d-a e) (b B d-4 A b e+3 a B e)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{4 b^3 e}\\ &=-\frac{(b B d-4 A b e+3 a B e) \sqrt{a+b x} \sqrt{d+e x}}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e}-\frac{(b d-a e) (b B d-4 A b e+3 a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.315114, size = 135, normalized size = 0.96 \[ \frac{\sqrt{d+e x} \left (\sqrt{e} \sqrt{a+b x} (-3 a B e+4 A b e+b B (d+2 e x))-\frac{\sqrt{b d-a e} (3 a B e-4 A b e+b B d) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{\sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{4 b^2 e^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]

[Out]

(Sqrt[d + e*x]*(Sqrt[e]*Sqrt[a + b*x]*(4*A*b*e - 3*a*B*e + b*B*(d + 2*e*x)) - (Sqrt[b*d - a*e]*(b*B*d - 4*A*b*
e + 3*a*B*e)*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]])/Sqrt[(b*(d + e*x))/(b*d - a*e)]))/(4*b^2*e^(3/2
))

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 375, normalized size = 2.7 \begin{align*} -{\frac{1}{8\,{b}^{2}e}\sqrt{ex+d}\sqrt{bx+a} \left ( 4\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ab{e}^{2}-4\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}de-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}+2\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) abde+B\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ){b}^{2}{d}^{2}-4\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xbe-8\,A\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }be+6\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae-2\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }bd \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(4*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/
2))*a*b*e^2-4*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d*e-3*B*ln(1/2
*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e^2+2*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(
e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d*e+B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2
)+a*e+b*d)/(b*e)^(1/2))*b^2*d^2-4*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*x*b*e-8*A*(b*e)^(1/2)*((b*x+a)*(e*x+d)
)^(1/2)*b*e+6*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*e-2*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b*d)/((b*x+a)*
(e*x+d))^(1/2)/b^2/e/(b*e)^(1/2)

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.42935, size = 840, normalized size = 6. \begin{align*} \left [\frac{{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt{b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \,{\left (2 \, b e x + b d + a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \,{\left (2 \, B b^{2} e^{2} x + B b^{2} d e -{\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{16 \, b^{3} e^{2}}, \frac{{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt{-b e} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \,{\left (2 \, B b^{2} e^{2} x + B b^{2} d e -{\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{8 \, b^{3} e^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*((B*b^2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a^2 - 4*A*a*b)*e^2)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 +
6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x
) + 4*(2*B*b^2*e^2*x + B*b^2*d*e - (3*B*a*b - 4*A*b^2)*e^2)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^2), 1/8*((B*b^
2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a^2 - 4*A*a*b)*e^2)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b
*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(2*B*b^2*e^2*x + B*b^2*d*
e - (3*B*a*b - 4*A*b^2)*e^2)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e^2)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.6189, size = 328, normalized size = 2.34 \begin{align*} -\frac{\frac{48 \,{\left (\frac{{\left (b^{2} d - a b e\right )} e^{\left (-\frac{1}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}\right )} A{\left | b \right |}}{b^{2}} - \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e^{\left (-2\right )}}{b^{4}} + \frac{{\left (b d e - 5 \, a e^{2}\right )} e^{\left (-4\right )}}{b^{4}}\right )} + \frac{{\left (b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{7}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{7}{2}}}\right )} B{\left | b \right |}}{b^{3}}}{48 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(48*((b^2*d - a*b*e)*e^(-1/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*
e)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a))*A*abs(b)/b^2 - (sqrt(b^2*d + (b*x + a)*b*e -
a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - 5*a*e^2)*e^(-4)/b^4) + (b^2*d^2 + 2*a*b*d*e - 3*a^2*e
^2)*e^(-7/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*abs(b)/
b^3)/b