3.2203 \(\int \frac{\sqrt{a+b x} (A+B x)}{\sqrt{d+e x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.105331, 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} \[ \frac{(b d-a e) (a B e-4 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{3/2} e^{5/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (a B e-4 A b e+3 b B d)}{4 b e^2}+\frac{B (a+b x)^{3/2} \sqrt{d+e x}}{2 b e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.644953, size = 185, normalized size = 1.32 \[ \frac{\sqrt{d+e x} \left (2 B e^2 (a+b x)^2-\frac{(a B e-4 A b e+3 b B d) \left (e (a+b x) \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}-\sqrt{e} \sqrt{a+b x} (b d-a e) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{\sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}}\right )}{4 b e^3 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.019, size = 376, normalized size = 2.7 \begin{align*}{\frac{1}{8\,{e}^{2}b}\sqrt{bx+a}\sqrt{ex+d} \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+4\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xbe-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 ){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+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 ){b}^{2}{d}^{2}+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 ) }bd+2\,B\sqrt{be}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae \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)*(b*x+a)^(1/2)/(e*x+d)^(1/2),x)

[Out]

1/8*(b*x+a)^(1/2)*(e*x+d)^(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+4*B*(b*e)^(
1/2)*((b*x+a)*(e*x+d))^(1/2)*x*b*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
))*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+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))*b^2*d^2+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)*b*d+2*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*e)/((b*x+a)*(
e*x+d))^(1/2)/e^2/b/(b*e)^(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)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.20501, size = 841, normalized size = 6.01 \begin{align*} \left [\frac{{\left (3 \, B b^{2} d^{2} - 2 \,{\left (B a b + 2 \, A b^{2}\right )} d e -{\left (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 - 3 \, B b^{2} d e +{\left (B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{16 \, b^{2} e^{3}}, -\frac{{\left (3 \, B b^{2} d^{2} - 2 \,{\left (B a b + 2 \, A b^{2}\right )} d e -{\left (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 - 3 \, B b^{2} d e +{\left (B a b + 4 \, A b^{2}\right )} e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{8 \, b^{2} e^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 1.44115, size = 236, normalized size = 1.69 \begin{align*} \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 )} B e^{\left (-1\right )}}{b^{2}} - \frac{{\left (3 \, B b^{3} d e + B a b^{2} e^{2} - 4 \, A b^{3} e^{2}\right )} e^{\left (-3\right )}}{b^{4}}\right )} - \frac{{\left (3 \, B b^{2} d^{2} - 2 \, B a b d e - 4 \, A b^{2} d e - B a^{2} e^{2} + 4 \, A a b e^{2}\right )} e^{\left (-\frac{5}{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{3}{2}}}\right )} b}{4 \,{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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