Integrand size = 22, antiderivative size = 112 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {2 c^2 \sqrt {a+b x}}{d^2 (b c-a d) \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^2}-\frac {(3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2}} \]
-(a*d+3*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(3/2)/ d^(5/2)+2*c^2*(b*x+a)^(1/2)/d^2/(-a*d+b*c)/(d*x+c)^(1/2)+(b*x+a)^(1/2)*(d* x+c)^(1/2)/b/d^2
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\sqrt {b} \sqrt {d} \sqrt {a+b x} (a d (c+d x)-b c (3 c+d x))+\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{5/2} (-b c+a d) \sqrt {c+d x}} \]
(Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*(a*d*(c + d*x) - b*c*(3*c + d*x)) + (3*b^2* c^2 - 2*a*b*c*d - a^2*d^2)*Sqrt[c + d*x]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/( Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*d^(5/2)*(-(b*c) + a*d)*Sqrt[c + d*x])
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 27, 90, 66, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}-\frac {2 \int \frac {(b c-a d) (c-d x)}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{d^2 (b c-a d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}-\frac {\int \frac {c-d x}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d^2}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(a d+3 b c) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}}{d^2}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(a d+3 b c) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}}{d^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{d^2 \sqrt {c+d x} (b c-a d)}-\frac {\frac {(a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}}{d^2}\) |
(2*c^2*Sqrt[a + b*x])/(d^2*(b*c - a*d)*Sqrt[c + d*x]) - (-((Sqrt[a + b*x]* Sqrt[c + d*x])/b) + ((3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b ]*Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d]))/d^2
3.8.42.3.1 Defintions of rubi rules used
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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. \(438\) vs. \(2(92)=184\).
Time = 0.58 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.92
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{3} x +2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{2} x -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d x +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c \,d^{2}+2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d -3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3}-2 a \,d^{2} x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 b c d x \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}-2 a c d \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+6 b \,c^{2} \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\right )}{2 \sqrt {b d}\, b \left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} \sqrt {d x +c}}\) | \(439\) |
-1/2*(b*x+a)^(1/2)*(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^2*d^3*x+2*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*b*c*d^2*x-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))*b^2*c^2*d*x+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^2*c*d^ 2+2*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*b*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))*b^2*c^3-2*a*d^2*x*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+ 2*b*c*d*x*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-2*a*c*d*(b*d)^(1/2)*((b*x+a) *(d*x+c))^(1/2)+6*b*c^2*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/(b*d)^(1/2)/b /(a*d-b*c)/((b*x+a)*(d*x+c))^(1/2)/d^2/(d*x+c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (92) = 184\).
Time = 0.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 4.18 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{4 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}, \frac {{\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} c^{2} d - a b c d^{2} + {\left (b^{2} c d^{2} - a b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{3} c^{2} d^{3} - a b^{2} c d^{4} + {\left (b^{3} c d^{4} - a b^{2} d^{5}\right )} x\right )}}\right ] \]
[1/4*((3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*x)*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*(3*b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*sqr t(b*x + a)*sqrt(d*x + c))/(b^3*c^2*d^3 - a*b^2*c*d^4 + (b^3*c*d^4 - a*b^2* d^5)*x), 1/2*((3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b* c*d^2 - a^2*d^3)*x)*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*(3*b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(b^3*c^2*d^3 - a*b^2*c*d^4 + (b^3*c*d^4 - a*b^2*d^5)*x)]
\[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (92) = 184\).
Time = 0.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.72 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\frac {\sqrt {b x + a} {\left (\frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} {\left (b x + a\right )}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}} + \frac {3 \, b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}}{b^{3} c d^{3} {\left | b \right |} - a b^{2} d^{4} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {{\left (3 \, b c + a d\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2} {\left | b \right |}} \]
sqrt(b*x + a)*((b^3*c*d^2 - a*b^2*d^3)*(b*x + a)/(b^3*c*d^3*abs(b) - a*b^2 *d^4*abs(b)) + (3*b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3)/(b^3*c*d^3*abs( b) - a*b^2*d^4*abs(b)))/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) + (3*b*c + a*d )*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d))) /(sqrt(b*d)*d^2*abs(b))
Timed out. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx=\int \frac {x^2}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]