3.1393 \(\int \frac{(c+d x)^{3/2}}{a+b x} \, dx\)

Optimal. Leaf size=86 \[ \frac{2 \sqrt{c+d x} (b c-a d)}{b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0455595, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {50, 63, 208} \[ \frac{2 \sqrt{c+d x} (b c-a d)}{b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}}+\frac{2 (c+d x)^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)/(a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0716295, size = 77, normalized size = 0.9 \[ \frac{2 \sqrt{c+d x} (-3 a d+4 b c+b d x)}{3 b^2}-\frac{2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x}}{\sqrt{b c-a d}}\right )}{b^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.006, size = 167, normalized size = 1.9 \begin{align*}{\frac{2}{3\,b} \left ( dx+c \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ad\sqrt{dx+c}}{{b}^{2}}}+2\,{\frac{\sqrt{dx+c}c}{b}}+2\,{\frac{{a}^{2}{d}^{2}}{{b}^{2}\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }-4\,{\frac{acd}{b\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) }+2\,{\frac{{c}^{2}}{\sqrt{ \left ( ad-bc \right ) b}}\arctan \left ({\frac{b\sqrt{dx+c}}{\sqrt{ \left ( ad-bc \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8924, size = 424, normalized size = 4.93 \begin{align*} \left [-\frac{3 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x + 2 \, b c - a d + 2 \, \sqrt{d x + c} b \sqrt{\frac{b c - a d}{b}}}{b x + a}\right ) - 2 \,{\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt{d x + c}}{3 \, b^{2}}, -\frac{2 \,{\left (3 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) -{\left (b d x + 4 \, b c - 3 \, a d\right )} \sqrt{d x + c}\right )}}{3 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 10.7802, size = 82, normalized size = 0.95 \begin{align*} \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 b} + \frac{\sqrt{c + d x} \left (- 2 a d + 2 b c\right )}{b^{2}} + \frac{2 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt{c + d x}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{b^{3} \sqrt{\frac{a d - b c}{b}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c + d*x)**(3/2)/(3*b) + sqrt(c + d*x)*(-2*a*d + 2*b*c)/b**2 + 2*(a*d - b*c)**2*atan(sqrt(c + d*x)/sqrt((a*d
- b*c)/b))/(b**3*sqrt((a*d - b*c)/b))

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Giac [A]  time = 1.06591, size = 142, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{d x + c} b^{2} c - 3 \, \sqrt{d x + c} a b d\right )}}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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