3.7.10 \(\int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx\) [610]

3.7.10.1 Optimal result

Integrand size = 22, antiderivative size = 213 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\frac {1}{8} \left (8 a c-\frac {b c^2}{d}+\frac {a^2 d}{b}\right ) \sqrt {a+b x} \sqrt {c+d x}+\frac {(b c+a d) \sqrt {a+b x} (c+d x)^{3/2}}{4 d}+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-2 a^{3/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {(b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{3/2} d^{3/2}} \]

output

1/3*(b*x+a)^(3/2)*(d*x+c)^(3/2)-2*a^(3/2)*c^(3/2)*arctanh(c^(1/2)*(b*x+a)^ 
(1/2)/a^(1/2)/(d*x+c)^(1/2))-1/8*(a*d+b*c)*(a^2*d^2-10*a*b*c*d+b^2*c^2)*ar 
ctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/d^(3/2)+1/4*(a* 
d+b*c)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/d+1/8*(8*a*c-b*c^2/d+a^2*d/b)*(b*x+a)^( 
1/2)*(d*x+c)^(1/2)
 

3.7.10.2 Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (3 a^2 d^2+2 a b d (19 c+7 d x)+b^2 \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 b d}-2 a^{3/2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )-\frac {\left (b^3 c^3-9 a b^2 c^2 d-9 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{8 b^{3/2} d^{3/2}} \]

input

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

output

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3*a^2*d^2 + 2*a*b*d*(19*c + 7*d*x) + b^2*(3* 
c^2 + 14*c*d*x + 8*d^2*x^2)))/(24*b*d) - 2*a^(3/2)*c^(3/2)*ArcTanh[(Sqrt[a 
]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])] - ((b^3*c^3 - 9*a*b^2*c^2*d - 9* 
a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b 
*x])])/(8*b^(3/2)*d^(3/2))
 

3.7.10.3 Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {112, 27, 171, 27, 171, 27, 175, 66, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}-\frac {1}{3} \int -\frac {3 \sqrt {a+b x} \sqrt {c+d x} (2 a c+(b c+a d) x)}{2 x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {a+b x} \sqrt {c+d x} (2 a c+(b c+a d) x)}{x}dx+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {c+d x} \left (8 a^2 c d-\left (b^2 c^2-8 a b d c-a^2 d^2\right ) x\right )}{2 x \sqrt {a+b x}}dx}{2 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {c+d x} \left (8 a^2 c d-\left (b^2 c^2-8 a b d c-a^2 d^2\right ) x\right )}{x \sqrt {a+b x}}dx}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {16 a^2 b c^2 d-(b c+a d) \left (b^2 c^2-10 a b d c+a^2 d^2\right ) x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {16 a^2 b c^2 d-(b c+a d) \left (b^2 c^2-10 a b d c+a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{2} \left (\frac {\frac {16 a^2 b c^2 d \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-(a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {1}{2} \left (\frac {\frac {16 a^2 b c^2 d \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 (a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{2} \left (\frac {\frac {32 a^2 b c^2 d \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-2 (a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (\frac {\frac {-32 a^{3/2} b c^{3/2} d \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )-\frac {2 (a d+b c) \left (a^2 d^2-10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-a^2 d^2-8 a b c d+b^2 c^2\right )}{b}}{4 d}+\frac {\sqrt {a+b x} (c+d x)^{3/2} (a d+b c)}{2 d}\right )+\frac {1}{3} (a+b x)^{3/2} (c+d x)^{3/2}\)

input

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

output

((a + b*x)^(3/2)*(c + d*x)^(3/2))/3 + (((b*c + a*d)*Sqrt[a + b*x]*(c + d*x 
)^(3/2))/(2*d) + (-(((b^2*c^2 - 8*a*b*c*d - a^2*d^2)*Sqrt[a + b*x]*Sqrt[c 
+ d*x])/b) + (-32*a^(3/2)*b*c^(3/2)*d*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqr 
t[a]*Sqrt[c + d*x])] - (2*(b*c + a*d)*(b^2*c^2 - 10*a*b*c*d + a^2*d^2)*Arc 
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/ 
(2*b))/(4*d))/2
 

3.7.10.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 112

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

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[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] /; Fre 
eQ[{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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(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 221

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

3.7.10.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(169)=338\).

Time = 1.55 (sec) , antiderivative size = 503, normalized size of antiderivative = 2.36

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-16 b^{2} d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{3} d^{3}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} b c \,d^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a \,b^{2} c^{2} d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, b^{3} c^{3}+48 \sqrt {b d}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,d^{2} x -28 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c d x -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2}-76 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d -6 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2}\right )}{48 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b d}\)	\(503\)

input

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

output

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-16*b^2*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d 
*x+c))^(1/2)*(b*d)^(1/2)+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d) 
^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*a^3*d^3-27*ln(1/2*(2*b*d*x+2*((b* 
x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*(a*c)^(1/2)*a^2*b*c* 
d^2-27*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+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+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+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*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c^2*d- 
28*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*d^2*x-28*(b*d)^(1/2 
)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c*d*x-6*(b*d)^(1/2)*(a*c)^(1/2)* 
((b*x+a)*(d*x+c))^(1/2)*a^2*d^2-76*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c 
))^(1/2)*a*b*c*d-6*(b*d)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2 
)/(b*d)^(1/2)/(a*c)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/b/d
 

3.7.10.5 Fricas [A] (verification not implemented)

Time = 3.26 (sec) , antiderivative size = 1193, normalized size of antiderivative = 5.60 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\text {Too large to display} \]

input

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

output

[1/96*(48*sqrt(a*c)*a*b^2*c*d^2*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*(b^3*c^3 - 9*a*b^2*c^2*d - 9*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 + 3*b^3*c^2*d + 38*a*b^2*c*d^2 + 3* 
a^2*b*d^3 + 14*(b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^ 
2*d^2), 1/48*(24*sqrt(a*c)*a*b^2*c*d^2*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*(b^3*c^3 - 9*a*b^2*c^2*d - 9 
*a^2*b*c*d^2 + a^3*d^3)*sqrt(-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 + 3*b^3*c^2*d + 38*a*b^2*c*d^2 + 3*a^2*b*d^3 + 
14*(b^3*c*d^2 + a*b^2*d^3)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^2*d^2), 1/96 
*(96*sqrt(-a*c)*a*b^2*c*d^2*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*(b^3*c^3 - 9*a*b^2*c^2*d - 9*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)*sqr 
t(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 + 3*b^3*c^2*d + 38*a*b^2*c*d^2 + 3*a^2*b*d^3 + 14*(b^3*c*d^2 + a...
 

3.7.10.6 Sympy [F]

\[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\int \frac {\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}{x}\, dx \]

input

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

output

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

3.7.10.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

3.7.10.8 Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

3.7.10.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}}{x} \,d x \]

input

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

output

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