3.2204 \(\int \frac{\sqrt{a+b x} (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac{(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{5/2}}-\frac{2 (a+b x)^{3/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.119679, antiderivative size = 148, 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 = {78, 50, 63, 217, 206} \[ \frac{\sqrt{a+b x} \sqrt{d+e x} (-a B e-2 A b e+3 b B d)}{e^2 (b d-a e)}-\frac{(-a B e-2 A b e+3 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{\sqrt{b} e^{5/2}}-\frac{2 (a+b x)^{3/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.355673, size = 129, normalized size = 0.87 \[ \frac{\sqrt{a+b x} (-2 A e+3 B d+B e x)}{e^2 \sqrt{d+e x}}-\frac{\sqrt{d+e x} (-a B e-2 A b e+3 b B d) \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )}{e^{5/2} \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.022, size = 386, normalized size = 2.6 \begin{align*}{\frac{1}{2\,{e}^{2}}\sqrt{bx+a} \left ( 2\,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 ) xb{e}^{2}+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 ) xa{e}^{2}-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 ) xbde+2\,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 ) bde+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 ) ade-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{d}^{2}+2\,Bxe\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}-4\,Ae\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+6\,Bd\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}{\frac{1}{\sqrt{ex+d}}}} \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)^(3/2),x)

[Out]

1/2*(b*x+a)^(1/2)*(2*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))*x*b*e^2+B*l
n(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))*x*a*e^2-3*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))*x*b*d*e+2*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*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))*a*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*d^2+2*B*x*e*((b*x
+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4*A*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+6*B*d*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2))/(b*e)^(1/2)/((b*x+a)*(e*x+d))^(1/2)/e^2/(e*x+d)^(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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.37236, size = 829, normalized size = 5.6 \begin{align*} \left [-\frac{{\left (3 \, B b d^{2} -{\left (B a + 2 \, A b\right )} d e +{\left (3 \, B b d e -{\left (B a + 2 \, A b\right )} e^{2}\right )} x\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 (B b e^{2} x + 3 \, B b d e - 2 \, A b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{4 \,{\left (b e^{4} x + b d e^{3}\right )}}, \frac{{\left (3 \, B b d^{2} -{\left (B a + 2 \, A b\right )} d e +{\left (3 \, B b d e -{\left (B a + 2 \, A b\right )} e^{2}\right )} x\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 (B b e^{2} x + 3 \, B b d e - 2 \, A b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d}}{2 \,{\left (b e^{4} x + b d e^{3}\right )}}\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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \sqrt{a + b x}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \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)**(3/2),x)

[Out]

Integral((A + B*x)*sqrt(a + b*x)/(d + e*x)**(3/2), x)

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Giac [A]  time = 1.88979, size = 274, normalized size = 1.85 \begin{align*} \frac{{\left (3 \, B b d{\left | b \right |} - B a{\left | b \right |} e - 2 \, A b{\left | b \right |} e\right )} \sqrt{b} e^{\frac{1}{2}} \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 )}{32 \,{\left (b^{6} d e^{4} - a b^{5} e^{5}\right )}} + \frac{{\left (\frac{{\left (b x + a\right )} B b{\left | b \right |} e^{2}}{b^{6} d e^{4} - a b^{5} e^{5}} + \frac{3 \, B b^{2} d{\left | b \right |} e - B a b{\left | b \right |} e^{2} - 2 \, A b^{2}{\left | b \right |} e^{2}}{b^{6} d e^{4} - a b^{5} e^{5}}\right )} \sqrt{b x + a}}{32 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} \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)^(3/2),x, algorithm="giac")

[Out]

1/32*(3*B*b*d*abs(b) - B*a*abs(b)*e - 2*A*b*abs(b)*e)*sqrt(b)*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)))/(b^6*d*e^4 - a*b^5*e^5) + 1/32*((b*x + a)*B*b*abs(b)*e^2/(b^6*d*e^4 - a
*b^5*e^5) + (3*B*b^2*d*abs(b)*e - B*a*b*abs(b)*e^2 - 2*A*b^2*abs(b)*e^2)/(b^6*d*e^4 - a*b^5*e^5))*sqrt(b*x + a
)/sqrt(b^2*d + (b*x + a)*b*e - a*b*e)