3.569 \(\int \frac{\sqrt{a+b x} (c+d x)^{5/2}}{x} \, dx\)

Optimal. Leaf size=219 \[ \frac{\left (-5 a^2 b c d^2+a^3 d^3+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d) (a d+b c)}{8 b^2}-2 \sqrt{a} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 b} \]

[Out]

((5*b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^2) + ((5*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)
)/(12*b) + (Sqrt[a + b*x]*(c + d*x)^(5/2))/3 - 2*Sqrt[a]*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt
[c + d*x])] + ((5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(5/2)*Sqrt[d])

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Rubi [A]  time = 0.222086, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {101, 154, 157, 63, 217, 206, 93, 208} \[ \frac{\left (-5 a^2 b c d^2+a^3 d^3+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} \sqrt{d}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d) (a d+b c)}{8 b^2}-2 \sqrt{a} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}+\frac{\sqrt{a+b x} (c+d x)^{3/2} (a d+5 b c)}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x,x]

[Out]

((5*b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*b^2) + ((5*b*c + a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)
)/(12*b) + (Sqrt[a + b*x]*(c + d*x)^(5/2))/3 - 2*Sqrt[a]*c^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt
[c + d*x])] + ((5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]
*Sqrt[c + d*x])])/(8*b^(5/2)*Sqrt[d])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[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)^{5/2}}{x} \, dx &=\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}-\frac{1}{3} \int \frac{(c+d x)^{3/2} \left (-3 a c+\frac{1}{2} (-5 b c-a d) x\right )}{x \sqrt{a+b x}} \, dx\\ &=\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}-\frac{\int \frac{\sqrt{c+d x} \left (-6 a b c^2-\frac{3}{4} (5 b c-a d) (b c+a d) x\right )}{x \sqrt{a+b x}} \, dx}{6 b}\\ &=\frac{(5 b c-a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^2}+\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}-\frac{\int \frac{-6 a b^2 c^3-\frac{3}{8} \left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) x}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{6 b^2}\\ &=\frac{(5 b c-a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^2}+\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}+\left (a c^3\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx+\frac{\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 b^2}\\ &=\frac{(5 b c-a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^2}+\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}+\left (2 a c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )+\frac{\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b^3}\\ &=\frac{(5 b c-a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^2}+\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}-2 \sqrt{a} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 b^3}\\ &=\frac{(5 b c-a d) (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{8 b^2}+\frac{(5 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 b}+\frac{1}{3} \sqrt{a+b x} (c+d x)^{5/2}-2 \sqrt{a} c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\frac{\left (5 b^3 c^3+15 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{5/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.899742, size = 274, normalized size = 1.25 \[ \frac{\sqrt{c+d x} \left (-\frac{b \sqrt{d} \left (\sqrt{a+b x} (c+d x) \left (-3 a^2 d^2+2 a b d (7 c+d x)+b^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )\right )-48 \sqrt{a} b^2 c^{5/2} \sqrt{c+d x} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )\right )}{\sqrt{\frac{b (c+d x)}{b c-a d}}}-3 \sqrt{b c-a d} \left (-5 a^2 b c d^2+a^3 d^3+15 a b^2 c^2 d+5 b^3 c^3\right ) \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b c-a d}}\right )\right )}{24 b^2 \sqrt{d} (a d-b c) \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[a + b*x]*(c + d*x)^(5/2))/x,x]

[Out]

(Sqrt[c + d*x]*(-3*Sqrt[b*c - a*d]*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*ArcSinh[(Sqrt[d]*Sqr
t[a + b*x])/Sqrt[b*c - a*d]] - (b*Sqrt[d]*(Sqrt[a + b*x]*(c + d*x)*(-3*a^2*d^2 + 2*a*b*d*(7*c + d*x) + b^2*(33
*c^2 + 26*c*d*x + 8*d^2*x^2)) - 48*Sqrt[a]*b^2*c^(5/2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*
Sqrt[c + d*x])]))/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(24*b^2*Sqrt[d]*(-(b*c) + a*d)*Sqrt[(b*(c + d*x))/(b*c - a
*d)])

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Maple [B]  time = 0.013, size = 583, normalized size = 2.7 \begin{align*}{\frac{1}{48\,{b}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 16\,{x}^{2}{b}^{2}{d}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\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 ) \sqrt{ac}{a}^{3}{d}^{3}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{a}^{2}bc{d}^{2}+45\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}a{b}^{2}{c}^{2}d+15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) \sqrt{ac}{b}^{3}{c}^{3}-48\,\sqrt{bd}\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) a{b}^{2}{c}^{3}+4\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}xab{d}^{2}+52\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}x{b}^{2}cd-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{a}^{2}{d}^{2}+28\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}abcd+66\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}\sqrt{ac}{b}^{2}{c}^{2} \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)^(5/2)*(b*x+a)^(1/2)/x,x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(16*x^2*b^2*d^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(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))*(a*c)^(1/2)*a^3*d^3-15*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))*(a*c)^(1/2)*a^2*b*c*d^2+45*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))*(a*c)^(1/2)*a*b^2*c^2*d+15*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))*(a*c)^(1/2)*b^3*c^3-48*(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)*a*b^2*c^3+4*(b*d*x^2+a*d*x+b*c*x+
a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x*a*b*d^2+52*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*x*b^2*
c*d-6*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*a^2*d^2+28*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)
^(1/2)*(a*c)^(1/2)*a*b*c*d+66*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*(a*c)^(1/2)*b^2*c^2)/(b*d*x^2+a*d*x+
b*c*x+a*c)^(1/2)/b^2/(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)^(5/2)*(b*x+a)^(1/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 49.3773, size = 2732, normalized size = 12.47 \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)^(5/2)*(b*x+a)^(1/2)/x,x, algorithm="fricas")

[Out]

[1/96*(48*sqrt(a*c)*b^3*c^2*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)
*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2
*b*c*d^2 + a^3*d^3)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt
(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(8*b^3*d^3*x^2 + 33*b^3*c^2*d + 14*a*b^2*c*d^
2 - 3*a^2*b*d^3 + 2*(13*b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d), 1/48*(24*sqrt(a*c)*b^3
*c^2*d*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a
)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*sqr
t(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*
c*d + a*b*d^2)*x)) + 2*(8*b^3*d^3*x^2 + 33*b^3*c^2*d + 14*a*b^2*c*d^2 - 3*a^2*b*d^3 + 2*(13*b^3*c*d^2 + a*b^2*
d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d), 1/96*(96*sqrt(-a*c)*b^3*c^2*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)
*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 3*(5*b^3*c^3 + 15*a
*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*sqrt(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x
+ b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(8*b^3*d^3*x^2 + 33*b^3*c^2
*d + 14*a*b^2*c*d^2 - 3*a^2*b*d^3 + 2*(13*b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^3*d), 1/48
*(48*sqrt(-a*c)*b^3*c^2*d*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c)/(a*b*c*d*x
^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*sqrt(-b*d)*a
rctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b
*d^2)*x)) + 2*(8*b^3*d^3*x^2 + 33*b^3*c^2*d + 14*a*b^2*c*d^2 - 3*a^2*b*d^3 + 2*(13*b^3*c*d^2 + a*b^2*d^3)*x)*s
qrt(b*x + a)*sqrt(d*x + c))/(b^3*d)]

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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{5}{2}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(5/2)*(b*x+a)**(1/2)/x,x)

[Out]

Integral(sqrt(a + b*x)*(c + d*x)**(5/2)/x, x)

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Giac [A]  time = 1.82513, size = 448, normalized size = 2.05 \begin{align*} -\frac{2 \, \sqrt{b d} a c^{3}{\left | b \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{1}{24} \, \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} d^{2}{\left | b \right |}}{b^{4}} + \frac{13 \, b^{9} c d^{5}{\left | b \right |} - 7 \, a b^{8} d^{6}{\left | b \right |}}{b^{12} d^{4}}\right )} + \frac{3 \,{\left (11 \, b^{10} c^{2} d^{4}{\left | b \right |} - 4 \, a b^{9} c d^{5}{\left | b \right |} + a^{2} b^{8} d^{6}{\left | b \right |}\right )}}{b^{12} d^{4}}\right )} - \frac{{\left (5 \, \sqrt{b d} b^{3} c^{3}{\left | b \right |} + 15 \, \sqrt{b d} a b^{2} c^{2} d{\left | b \right |} - 5 \, \sqrt{b d} a^{2} b c d^{2}{\left | b \right |} + \sqrt{b d} a^{3} d^{3}{\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 )}{16 \, b^{4} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(5/2)*(b*x+a)^(1/2)/x,x, algorithm="giac")

[Out]

-2*sqrt(b*d)*a*c^3*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) + 1/24*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2
*(b*x + a)*(4*(b*x + a)*d^2*abs(b)/b^4 + (13*b^9*c*d^5*abs(b) - 7*a*b^8*d^6*abs(b))/(b^12*d^4)) + 3*(11*b^10*c
^2*d^4*abs(b) - 4*a*b^9*c*d^5*abs(b) + a^2*b^8*d^6*abs(b))/(b^12*d^4)) - 1/16*(5*sqrt(b*d)*b^3*c^3*abs(b) + 15
*sqrt(b*d)*a*b^2*c^2*d*abs(b) - 5*sqrt(b*d)*a^2*b*c*d^2*abs(b) + sqrt(b*d)*a^3*d^3*abs(b))*log((sqrt(b*d)*sqrt
(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(b^4*d)