3.1761 \(\int \frac{(A+B x) \sqrt{d+e x}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{e (-3 a B e-A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}}-\frac{(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.128698, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 47, 63, 208} \[ -\frac{\sqrt{d+e x} (-3 a B e-A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}-\frac{e (-3 a B e-A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} (b d-a e)^{3/2}}-\frac{(d+e x)^{3/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*b*B*d - A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(3/2))/(2*
b*(b*d - a*e)*(a + b*x)^2) - (e*(4*b*B*d - A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/
(4*b^(5/2)*(b*d - a*e)^(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 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 208

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

Rubi steps

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

Mathematica [A]  time = 0.311526, size = 158, normalized size = 1.01 \[ \frac{\frac{(a+b x) (-3 a B e-A b e+4 b B d) \left (\sqrt{b} e (a+b x) \sqrt{d+e x} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )-b (d+e x) \sqrt{a e-b d}\right )}{\sqrt{a e-b d}}-2 b^2 (d+e x)^2 (A b-a B)}{4 b^3 (a+b x)^2 \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.013, size = 339, normalized size = 2.2 \begin{align*}{\frac{A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{eBd}{ \left ( bxe+ae \right ) ^{2} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{A{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2}b}\sqrt{ex+d}}-{\frac{3\,Ba{e}^{2}}{4\, \left ( bxe+ae \right ) ^{2}{b}^{2}}\sqrt{ex+d}}+{\frac{eBd}{ \left ( bxe+ae \right ) ^{2}b}\sqrt{ex+d}}+{\frac{A{e}^{2}}{ \left ( 4\,ae-4\,bd \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{3\,Ba{e}^{2}}{ \left ( 4\,ae-4\,bd \right ){b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{eBd}{ \left ( ae-bd \right ) b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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)^3,x)

[Out]

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

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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)*(e*x+d)^(1/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.58357, size = 1481, normalized size = 9.43 \begin{align*} \left [\frac{{\left (4 \, B a^{2} b d e -{\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (2 \,{\left (B a b^{3} + A b^{4}\right )} d^{2} -{\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e +{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} +{\left (4 \, B b^{4} d^{2} -{\left (9 \, B a b^{3} - A b^{4}\right )} d e +{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} +{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \,{\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}, \frac{{\left (4 \, B a^{2} b d e -{\left (3 \, B a^{3} + A a^{2} b\right )} e^{2} +{\left (4 \, B b^{3} d e -{\left (3 \, B a b^{2} + A b^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (4 \, B a b^{2} d e -{\left (3 \, B a^{2} b + A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (2 \,{\left (B a b^{3} + A b^{4}\right )} d^{2} -{\left (5 \, B a^{2} b^{2} + 3 \, A a b^{3}\right )} d e +{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} e^{2} +{\left (4 \, B b^{4} d^{2} -{\left (9 \, B a b^{3} - A b^{4}\right )} d e +{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (a^{2} b^{5} d^{2} - 2 \, a^{3} b^{4} d e + a^{4} b^{3} e^{2} +{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} x^{2} + 2 \,{\left (a b^{6} d^{2} - 2 \, a^{2} b^{5} d e + a^{3} b^{4} e^{2}\right )} x\right )}}\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)^3,x, algorithm="fricas")

[Out]

[1/8*((4*B*a^2*b*d*e - (3*B*a^3 + A*a^2*b)*e^2 + (4*B*b^3*d*e - (3*B*a*b^2 + A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*
e - (3*B*a^2*b + A*a*b^2)*e^2)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*
x + d))/(b*x + a)) - 2*(2*(B*a*b^3 + A*b^4)*d^2 - (5*B*a^2*b^2 + 3*A*a*b^3)*d*e + (3*B*a^3*b + A*a^2*b^2)*e^2
+ (4*B*b^4*d^2 - (9*B*a*b^3 - A*b^4)*d*e + (5*B*a^2*b^2 - A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^2 - 2*a^3
*b^4*d*e + a^4*b^3*e^2 + (b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*x^2 + 2*(a*b^6*d^2 - 2*a^2*b^5*d*e + a^3*b^4*e^
2)*x), 1/4*((4*B*a^2*b*d*e - (3*B*a^3 + A*a^2*b)*e^2 + (4*B*b^3*d*e - (3*B*a*b^2 + A*b^3)*e^2)*x^2 + 2*(4*B*a*
b^2*d*e - (3*B*a^2*b + A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x
+ b*d)) - (2*(B*a*b^3 + A*b^4)*d^2 - (5*B*a^2*b^2 + 3*A*a*b^3)*d*e + (3*B*a^3*b + A*a^2*b^2)*e^2 + (4*B*b^4*d^
2 - (9*B*a*b^3 - A*b^4)*d*e + (5*B*a^2*b^2 - A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^2 - 2*a^3*b^4*d*e + a^
4*b^3*e^2 + (b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^2)*x^2 + 2*(a*b^6*d^2 - 2*a^2*b^5*d*e + a^3*b^4*e^2)*x)]

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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)*(e*x+d)**(1/2)/(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.11847, size = 331, normalized size = 2.11 \begin{align*} \frac{{\left (4 \, B b d e - 3 \, B a e^{2} - A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d - a b^{2} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} +{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} + 7 \, \sqrt{x e + d} B a b d e^{2} + \sqrt{x e + d} A b^{2} d e^{2} - 3 \, \sqrt{x e + d} B a^{2} e^{3} - \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d - a b^{2} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

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