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

Optimal. Leaf size=144 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{c} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.110609, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {97, 154, 157, 63, 217, 206, 93, 208} \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{c} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, 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])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{a+b x} (c+d x)^{3/2}}{x^2} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+\int \frac{\sqrt{c+d x} \left (\frac{1}{2} (b c+3 a d)+2 b d x\right )}{x \sqrt{a+b x}} \, dx\\ &=2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+\frac{\int \frac{\frac{1}{2} b c (b c+3 a d)+\frac{1}{2} b d (3 b c+a d) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{b}\\ &=2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+\frac{1}{2} (d (3 b c+a d)) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{1}{2} (c (b c+3 a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx\\ &=2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}+\frac{(d (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{b}+(c (b c+3 a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )\\ &=2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}-\frac{\sqrt{c} (b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{(d (3 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{b}\\ &=2 d \sqrt{a+b x} \sqrt{c+d x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{x}-\frac{\sqrt{c} (b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (3 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.904734, size = 174, normalized size = 1.21 \[ \frac{\frac{\sqrt{a+b x} \left (d^2 x^2-c^2\right )}{x}+\frac{\sqrt{d} \sqrt{b c-a d} (a d+3 b c) \sqrt{\frac{b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )}{b}-\frac{\sqrt{c} \sqrt{c+d x} (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}}{\sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a + b*x]*(-c^2 + d^2*x^2))/x + (Sqrt[d]*Sqrt[b*c - a*d]*(3*b*c + a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*A
rcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/b - (Sqrt[c]*(b*c + 3*a*d)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqr
t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a])/Sqrt[c + d*x]

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Maple [B]  time = 0.016, size = 347, normalized size = 2.4 \begin{align*}{\frac{1}{2\,x}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) xa{d}^{2}\sqrt{ac}+3\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xbcd\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xacd\sqrt{bd}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ) xb{c}^{2}\sqrt{bd}+2\,xd\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}-2\,c\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.93692, size = 2026, normalized size = 14.07 \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)^(3/2)*(b*x+a)^(1/2)/x^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.71157, size = 714, normalized size = 4.96 \begin{align*} \frac{\frac{2 \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a} d{\left | b \right |}}{b} - \frac{{\left (3 \, \sqrt{b d} b c{\left | b \right |} + \sqrt{b d} a d{\left | b \right |}\right )} \log \left ({\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{b} - \frac{2 \,{\left (\sqrt{b d} b^{2} c^{2}{\left | b \right |} + 3 \, \sqrt{b d} a b c d{\left | b \right |}\right )} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} b} - \frac{4 \,{\left (\sqrt{b d} b^{4} c^{3}{\left | b \right |} - 2 \, \sqrt{b d} a b^{3} c^{2} d{\left | b \right |} + \sqrt{b d} a^{2} b^{2} c d^{2}{\left | b \right |} - \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c^{2}{\left | b \right |} - \sqrt{b d}{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b c d{\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b d +{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4}}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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