Integrand size = 22, antiderivative size = 126 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {2 c^2 \sqrt {a+b x}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (2 b c-3 a d) \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \]
2*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(5/2)/b^(1/2)+2/3 *c^2*(b*x+a)^(1/2)/d^2/(-a*d+b*c)/(d*x+c)^(3/2)-4/3*c*(-3*a*d+2*b*c)*(b*x+ a)^(1/2)/d^2/(-a*d+b*c)^2/(d*x+c)^(1/2)
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c \sqrt {a+b x} \left (3 b c-6 a d+\frac {c d (a+b x)}{c+d x}\right )}{3 d^2 (-b c+a d)^2 \sqrt {c+d x}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{5/2}} \]
(-2*c*Sqrt[a + b*x]*(3*b*c - 6*a*d + (c*d*(a + b*x))/(c + d*x)))/(3*d^2*(- (b*c) + a*d)^2*Sqrt[c + d*x]) + (2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b ]*Sqrt[c + d*x])])/(Sqrt[b]*d^(5/2))
Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 27, 87, 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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {2 \int \frac {c (b c-3 a d)-3 d (b c-a d) x}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d^2 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\int \frac {c (b c-3 a d)-3 d (b c-a d) x}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d^2 (b c-a d)}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c \sqrt {a+b x} (2 b c-3 a d)}{\sqrt {c+d x} (b c-a d)}-3 (b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{3 d^2 (b c-a d)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c \sqrt {a+b x} (2 b c-3 a d)}{\sqrt {c+d x} (b c-a d)}-6 (b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{3 d^2 (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 c^2 \sqrt {a+b x}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {\frac {4 c \sqrt {a+b x} (2 b c-3 a d)}{\sqrt {c+d x} (b c-a d)}-\frac {6 (b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {d}}}{3 d^2 (b c-a d)}\) |
(2*c^2*Sqrt[a + b*x])/(3*d^2*(b*c - a*d)*(c + d*x)^(3/2)) - ((4*c*(2*b*c - 3*a*d)*Sqrt[a + b*x])/((b*c - a*d)*Sqrt[c + d*x]) - (6*(b*c - a*d)*ArcTan h[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/(3* d^2*(b*c - a*d))
3.8.50.3.1 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*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(603\) vs. \(2(102)=204\).
Time = 0.58 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.79
| method | result | size |
| default | \(\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} d^{4} x^{2}-6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b c \,d^{3} x^{2}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{2} d^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c \,d^{3} x -12 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{2} d^{2} x +6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{3} d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} c^{2} d^{2}-6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a b \,c^{3} d +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{2} c^{4}+12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a c \,d^{2} x -8 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{2} d x +10 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a \,c^{2} d -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, b \,c^{3}\right )}{3 \sqrt {b d}\, \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{2} \left (d x +c \right )^{\frac {3}{2}}}\) | \(604\) |
1/3*(b*x+a)^(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^2*d^4*x^2-6*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^3*x^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))*b^2*c^2*d^2*x^2+6 *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^3*x-12*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^2*x+6*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*d*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))*a^2*c^2*d^2-6*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^3*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^4+12*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*c*d^2*x-8*((b*x +a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b*c^2*d*x+10*((b*x+a)*(d*x+c))^(1/2)*(b*d)^ (1/2)*a*c^2*d-6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b*c^3)/(b*d)^(1/2)/(a* d-b*c)^2/((b*x+a)*(d*x+c))^(1/2)/d^2/(d*x+c)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (102) = 204\).
Time = 0.36 (sec) , antiderivative size = 670, normalized size of antiderivative = 5.32 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} + {\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}, -\frac {3 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (3 \, b^{2} c^{3} d - 5 \, a b c^{2} d^{2} + 2 \, {\left (2 \, b^{2} c^{2} d^{2} - 3 \, a b c d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (b^{3} c^{4} d^{3} - 2 \, a b^{2} c^{3} d^{4} + a^{2} b c^{2} d^{5} + {\left (b^{3} c^{2} d^{5} - 2 \, a b^{2} c d^{6} + a^{2} b d^{7}\right )} x^{2} + 2 \, {\left (b^{3} c^{3} d^{4} - 2 \, a b^{2} c^{2} d^{5} + a^{2} b c d^{6}\right )} x\right )}}\right ] \]
[1/6*(3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)*sqrt(b*d)*lo g(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^3*d - 5*a*b*c^2*d^2 + 2*(2*b^2*c^2*d^2 - 3*a*b*c*d^3)*x)*sqrt(b*x + a) *sqrt(d*x + c))/(b^3*c^4*d^3 - 2*a*b^2*c^3*d^4 + a^2*b*c^2*d^5 + (b^3*c^2* d^5 - 2*a*b^2*c*d^6 + a^2*b*d^7)*x^2 + 2*(b^3*c^3*d^4 - 2*a*b^2*c^2*d^5 + a^2*b*c*d^6)*x), -1/3*(3*(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2 + (b^2*c^2*d ^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*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^3*d - 5*a*b*c^2*d^2 + 2*(2*b^2*c^2*d^2 - 3*a*b*c*d^3)*x)*sqrt(b*x + a)*s qrt(d*x + c))/(b^3*c^4*d^3 - 2*a*b^2*c^3*d^4 + a^2*b*c^2*d^5 + (b^3*c^2*d^ 5 - 2*a*b^2*c*d^6 + a^2*b*d^7)*x^2 + 2*(b^3*c^3*d^4 - 2*a*b^2*c^2*d^5 + a^ 2*b*c*d^6)*x)]
\[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{2}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 227 vs. \(2 (102) = 204\).
Time = 0.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.80 \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (2 \, b^{6} c^{2} d^{2} - 3 \, a b^{5} c d^{3}\right )} {\left (b x + a\right )}}{b^{4} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{3} c d^{4} {\left | b \right |} + a^{2} b^{2} d^{5} {\left | b \right |}} + \frac {3 \, {\left (b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 2 \, a^{2} b^{5} c d^{3}\right )}}{b^{4} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{3} c d^{4} {\left | b \right |} + a^{2} b^{2} d^{5} {\left | b \right |}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {2 \, b \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{2} {\left | b \right |}} \]
-2/3*sqrt(b*x + a)*(2*(2*b^6*c^2*d^2 - 3*a*b^5*c*d^3)*(b*x + a)/(b^4*c^2*d ^3*abs(b) - 2*a*b^3*c*d^4*abs(b) + a^2*b^2*d^5*abs(b)) + 3*(b^7*c^3*d - 3* a*b^6*c^2*d^2 + 2*a^2*b^5*c*d^3)/(b^4*c^2*d^3*abs(b) - 2*a*b^3*c*d^4*abs(b ) + a^2*b^2*d^5*abs(b)))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 2*b*log(a bs(-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. \[ \int \frac {x^2}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^2}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]