3.1383 \(\int \frac{\sqrt{c+d x}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=110 \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}-\frac{d \sqrt{c+d x}}{4 b (a+b x) (b c-a d)}-\frac{\sqrt{c+d x}}{2 b (a+b x)^2} \]

[Out]

-Sqrt[c + d*x]/(2*b*(a + b*x)^2) - (d*Sqrt[c + d*x])/(4*b*(b*c - a*d)*(a + b*x)) + (d^2*ArcTanh[(Sqrt[b]*Sqrt[
c + d*x])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0780258, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 51, 63, 208} \[ \frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}-\frac{d \sqrt{c+d x}}{4 b (a+b x) (b c-a d)}-\frac{\sqrt{c+d x}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(a + b*x)^3,x]

[Out]

-Sqrt[c + d*x]/(2*b*(a + b*x)^2) - (d*Sqrt[c + d*x])/(4*b*(b*c - a*d)*(a + b*x)) + (d^2*ArcTanh[(Sqrt[b]*Sqrt[
c + d*x])/Sqrt[b*c - a*d]])/(4*b^(3/2)*(b*c - a*d)^(3/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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{\sqrt{c+d x}}{(a+b x)^3} \, dx &=-\frac{\sqrt{c+d x}}{2 b (a+b x)^2}+\frac{d \int \frac{1}{(a+b x)^2 \sqrt{c+d x}} \, dx}{4 b}\\ &=-\frac{\sqrt{c+d x}}{2 b (a+b x)^2}-\frac{d \sqrt{c+d x}}{4 b (b c-a d) (a+b x)}-\frac{d^2 \int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx}{8 b (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{2 b (a+b x)^2}-\frac{d \sqrt{c+d x}}{4 b (b c-a d) (a+b x)}-\frac{d \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x}\right )}{4 b (b c-a d)}\\ &=-\frac{\sqrt{c+d x}}{2 b (a+b x)^2}-\frac{d \sqrt{c+d x}}{4 b (b c-a d) (a+b x)}+\frac{d^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{4 b^{3/2} (b c-a d)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0172315, size = 52, normalized size = 0.47 \[ \frac{2 d^2 (c+d x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};-\frac{b (c+d x)}{a d-b c}\right )}{3 (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^2*(c + d*x)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, -((b*(c + d*x))/(-(b*c) + a*d))])/(3*(-(b*c) + a*d)^3)

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 111, normalized size = 1. \begin{align*}{\frac{{d}^{2}}{4\, \left ( bdx+ad \right ) ^{2} \left ( ad-bc \right ) } \left ( dx+c \right ) ^{{\frac{3}{2}}}}-{\frac{{d}^{2}}{4\, \left ( bdx+ad \right ) ^{2}b}\sqrt{dx+c}}+{\frac{{d}^{2}}{ \left ( 4\,ad-4\,bc \right ) b}\arctan \left ({b\sqrt{dx+c}{\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ad-bc \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(b*x+a)^3,x)

[Out]

1/4*d^2/(b*d*x+a*d)^2/(a*d-b*c)*(d*x+c)^(3/2)-1/4*d^2/(b*d*x+a*d)^2/b*(d*x+c)^(1/2)+1/4*d^2/(a*d-b*c)/b/((a*d-
b*c)*b)^(1/2)*arctan(b*(d*x+c)^(1/2)/((a*d-b*c)*b)^(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((d*x+c)^(1/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.16923, size = 949, normalized size = 8.63 \begin{align*} \left [-\frac{{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{b^{2} c - a b d} \log \left (\frac{b d x + 2 \, b c - a d - 2 \, \sqrt{b^{2} c - a b d} \sqrt{d x + c}}{b x + a}\right ) + 2 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{8 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}, -\frac{{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d} \arctan \left (\frac{\sqrt{-b^{2} c + a b d} \sqrt{d x + c}}{b d x + b c}\right ) +{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{4 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} +{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{2} + 2 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[-1/8*((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b
*d)*sqrt(d*x + c))/(b*x + a)) + 2*(2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x)*sqrt(d*x + c
))/(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^2 + 2*(a*b^5*c^2 - 2*a
^2*b^4*c*d + a^3*b^3*d^2)*x), -1/4*((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^
2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) + (2*b^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2 + (b^3*c*d - a*b^2*d^2)*x)*sq
rt(d*x + c))/(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^2 + 2*(a*b^5
*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x)]

________________________________________________________________________________________

Sympy [B]  time = 92.7819, size = 1658, normalized size = 15.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*a**2*d**4*sqrt(c + d*x)/(8*a**4*b*d**4 - 16*a**3*b**2*c*d**3 + 16*a**3*b**2*d**4*x - 48*a**2*b**3*c*d**3*x
+ 8*a**2*b**3*d**2*(c + d*x)**2 + 16*a*b**4*c**3*d + 48*a*b**4*c**2*d**2*x - 16*a*b**4*c*d*(c + d*x)**2 - 8*b
**5*c**4 - 16*b**5*c**3*d*x + 8*b**5*c**2*(c + d*x)**2) + 20*a*c*d**3*sqrt(c + d*x)/(8*a**4*d**4 - 16*a**3*b*c
*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**
3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) - 6*a*
d**3*(c + d*x)**(3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2
*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*
b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) + 3*a*d**3*sqrt(-1/(b*(a*d - b*c)**5))*log(-a**3*d**3*sqrt(-1/(b*(a*
d - b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d - b*c)**5)) + b*
*3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) - 3*a*d**3*sqrt(-1/(b*(a*d - b*c)**5))*log(a**3*d**
3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**2*d*sqrt(-1/(b*(a*d
- b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/(8*b) - 10*b*c**2*d**2*sqrt(c + d*x)/(8*a
**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*
b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**
2*(c + d*x)**2) + 6*b*c*d**2*(c + d*x)**(3/2)/(8*a**4*d**4 - 16*a**3*b*c*d**3 + 16*a**3*b*d**4*x - 48*a**2*b**
2*c*d**3*x + 8*a**2*b**2*d**2*(c + d*x)**2 + 16*a*b**3*c**3*d + 48*a*b**3*c**2*d**2*x - 16*a*b**3*c*d*(c + d*x
)**2 - 8*b**4*c**4 - 16*b**4*c**3*d*x + 8*b**4*c**2*(c + d*x)**2) - 3*c*d**2*sqrt(-1/(b*(a*d - b*c)**5))*log(-
a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a*b**2*c**2*d*sqrt(-1/
(b*(a*d - b*c)**5)) + b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8 + 3*c*d**2*sqrt(-1/(b*(a*d - b*
c)**5))*log(a**3*d**3*sqrt(-1/(b*(a*d - b*c)**5)) - 3*a**2*b*c*d**2*sqrt(-1/(b*(a*d - b*c)**5)) + 3*a*b**2*c**
2*d*sqrt(-1/(b*(a*d - b*c)**5)) - b**3*c**3*sqrt(-1/(b*(a*d - b*c)**5)) + sqrt(c + d*x))/8 + 2*d**2*sqrt(c + d
*x)/(2*a**2*b*d**2 - 2*a*b**2*c*d + 2*a*b**2*d**2*x - 2*b**3*c*d*x) - d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(-a*
*2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) + 2*a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) - b**2*c**2*sqrt(-1/(b*(a*d - b*c)
**3)) + sqrt(c + d*x))/(2*b) + d**2*sqrt(-1/(b*(a*d - b*c)**3))*log(a**2*d**2*sqrt(-1/(b*(a*d - b*c)**3)) - 2*
a*b*c*d*sqrt(-1/(b*(a*d - b*c)**3)) + b**2*c**2*sqrt(-1/(b*(a*d - b*c)**3)) + sqrt(c + d*x))/(2*b)

________________________________________________________________________________________

Giac [A]  time = 1.07411, size = 170, normalized size = 1.55 \begin{align*} -\frac{d^{2} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \,{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}} - \frac{{\left (d x + c\right )}^{\frac{3}{2}} b d^{2} + \sqrt{d x + c} b c d^{2} - \sqrt{d x + c} a d^{3}}{4 \,{\left (b^{2} c - a b d\right )}{\left ({\left (d x + c\right )} b - b c + a d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/4*d^2*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((b^2*c - a*b*d)*sqrt(-b^2*c + a*b*d)) - 1/4*((d*x + c)^
(3/2)*b*d^2 + sqrt(d*x + c)*b*c*d^2 - sqrt(d*x + c)*a*d^3)/((b^2*c - a*b*d)*((d*x + c)*b - b*c + a*d)^2)