Integrand size = 22, antiderivative size = 119 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2}}+\frac {2 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2}} \] Output:
2*(-a*d+b*c)*(d*x+c)^(1/2)/a/b/(b*x+a)^(1/2)-2*c^(3/2)*arctanh(c^(1/2)*(b* x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(3/2)+2*d^(3/2)*arctanh(d^(1/2)*(b*x+a )^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)
Result contains complex when optimal does not.
Time = 3.41 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.62 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\frac {2 (b c-a d) \sqrt {c+d x}}{a b \sqrt {a+b x}}+\frac {2 i \sqrt {c} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{3/2} b}-\frac {2 i \sqrt {c} \left (-i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{a^{3/2} b}-\frac {4 d^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b^{3/2}} \] Input:
Integrate[(c + d*x)^(3/2)/(x*(a + b*x)^(3/2)),x]
Output:
(2*(b*c - a*d)*Sqrt[c + d*x])/(a*b*Sqrt[a + b*x]) + ((2*I)*Sqrt[c]*(I*Sqrt [a]*Sqrt[d] + Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sq rt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]) )])/(a^(3/2)*b) - ((2*I)*Sqrt[c]*((-I)*Sqrt[a]*Sqrt[d] + Sqrt[b*c - a*d])* Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]* Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(a^(3/2)*b) - (4*d^(3/2)*Ar cTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]) )])/b^(3/2)
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {109, 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.
\(\displaystyle \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {2 \int \frac {b c^2+a d^2 x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {b c^2+a d^2 x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
\(\Big \downarrow \) 175 |
\(\displaystyle \frac {b c^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+a d^2 \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {b c^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+2 a d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {2 b c^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+2 a d^2 \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {2 a d^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 b c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}}}{a b}+\frac {2 \sqrt {c+d x} (b c-a d)}{a b \sqrt {a+b x}}\) |
Input:
Int[(c + d*x)^(3/2)/(x*(a + b*x)^(3/2)),x]
Output:
(2*(b*c - a*d)*Sqrt[c + d*x])/(a*b*Sqrt[a + b*x]) + ((-2*b*c^(3/2)*ArcTanh [(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] + (2*a*d^(3/2)* ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[b])/(a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(91)=182\).
Time = 0.26 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x \sqrt {d b}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a b \,d^{2} x \sqrt {a c}-\sqrt {d b}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{2}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) \sqrt {a c}\, a^{2} d^{2}-2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a d +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, b c \right ) \sqrt {x d +c}}{\sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, \sqrt {b x +a}\, a b}\) | \(306\) |
Input:
int((d*x+c)^(3/2)/x/(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^2*c^2* x*(d*b)^(1/2)+ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b* c)/(d*b)^(1/2))*a*b*d^2*x*(a*c)^(1/2)-(d*b)^(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*b*c^2+ln(1/2*(2*b*d*x+2*((b*x+a) *(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*(a*c)^(1/2)*a^2*d^2-2*(( b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a*d+2*((b*x+a)*(d*x+c))^(1/2 )*(d*b)^(1/2)*(a*c)^(1/2)*b*c)*(d*x+c)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/(d*b) ^(1/2)/(a*c)^(1/2)/(b*x+a)^(1/2)/a/b
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (91) = 182\).
Time = 0.38 (sec) , antiderivative size = 956, normalized size of antiderivative = 8.03 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)/x/(b*x+a)^(3/2),x, algorithm="fricas")
Output:
[1/2*((a*b*d*x + a^2*d)*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^2*c*x + a*b*c)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(a*b^2*x + a^2*b), -1/2*(2*(a*b *d*x + a^2*d)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sq rt(d*x + c)*sqrt(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) - (b^2*c*x + a*b*c)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/ (a*b^2*x + a^2*b), 1/2*(2*(b^2*c*x + a*b*c)*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)) + (a*b*d*x + a^2*d)*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*(b*c - a*d)*sqrt(b* x + a)*sqrt(d*x + c))/(a*b^2*x + a^2*b), ((b^2*c*x + a*b*c)*sqrt(-c/a)*arc tan(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)) - (a*b*d*x + a^2*d)*sqrt(-d/b)*arcta n(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)/(b*d...
\[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{x \left (a + b x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x+c)**(3/2)/x/(b*x+a)**(3/2),x)
Output:
Integral((c + d*x)**(3/2)/(x*(a + b*x)**(3/2)), x)
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((d*x+c)^(3/2)/x/(b*x+a)^(3/2),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
Exception generated. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x+c)^(3/2)/x/(b*x+a)^(3/2),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
Timed out. \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{x\,{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:
int((c + d*x)^(3/2)/(x*(a + b*x)^(3/2)),x)
Output:
int((c + d*x)^(3/2)/(x*(a + b*x)^(3/2)), x)
Time = 0.27 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.35 \[ \int \frac {(c+d x)^{3/2}}{x (a+b x)^{3/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c +\sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c -\sqrt {c}\, \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b^{2} c +2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d -2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a^{2} d +2 \sqrt {d}\, \sqrt {b}\, \sqrt {b x +a}\, a b c -2 \sqrt {d x +c}\, a^{2} b d +2 \sqrt {d x +c}\, a \,b^{2} c}{\sqrt {b x +a}\, a^{2} b^{2}} \] Input:
int((d*x+c)^(3/2)/x/(b*x+a)^(3/2),x)
Output:
(sqrt(c)*sqrt(a)*sqrt(a + b*x)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt( a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c + sqrt(c)*sqrt(a)*sqrt(a + b*x)*log(sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*b**2*c - sqrt (c)*sqrt(a)*sqrt(a + b*x)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x ) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*b**2*c + 2*sqrt(d)*sqrt(b )*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a *d - b*c))*a**2*d - 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**2*d + 2*sqrt(d)*sqr t(b)*sqrt(a + b*x)*a*b*c - 2*sqrt(c + d*x)*a**2*b*d + 2*sqrt(c + d*x)*a*b* *2*c)/(sqrt(a + b*x)*a**2*b**2)