Integrand size = 38, antiderivative size = 108 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {6 d^2 \sqrt {a+b x}}{(b d-a e) \sqrt {d+e x}}+\frac {8 \sqrt {a+b x} \sqrt {d+e x}}{b}+\frac {8 (2 b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}} \]
8*(-a*e+2*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2))/b^(3/2 )/e^(1/2)+6*d^2*(b*x+a)^(1/2)/(-a*e+b*d)/(e*x+d)^(1/2)+8*(b*x+a)^(1/2)*(e* x+d)^(1/2)/b
Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {2 \left (\frac {\sqrt {b} \sqrt {a+b x} (-4 a e (d+e x)+b d (7 d+4 e x))}{\sqrt {d+e x}}+\frac {4 \left (2 b^2 d^2-3 a b d e+a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {e}}\right )}{b^{3/2} (b d-a e)} \]
(2*((Sqrt[b]*Sqrt[a + b*x]*(-4*a*e*(d + e*x) + b*d*(7*d + 4*e*x)))/Sqrt[d + e*x] + (4*(2*b^2*d^2 - 3*a*b*d*e + a^2*e^2)*ArcTanh[(Sqrt[e]*Sqrt[a + b* x])/(Sqrt[b]*Sqrt[d + e*x])])/Sqrt[e]))/(b^(3/2)*(b*d - a*e))
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1193, 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.
| \(\displaystyle \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1193 |
| \(\displaystyle \frac {2 \int \frac {2 (b d-a e) (3 d+2 e x)}{\sqrt {a+b x} \sqrt {d+e x}}dx}{b d-a e}+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {3 d+2 e x}{\sqrt {a+b x} \sqrt {d+e x}}dx+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle 4 \left (\frac {(2 b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {a+b x} \sqrt {d+e x}}{b}\right )+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle 4 \left (\frac {2 (2 b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {2 \sqrt {a+b x} \sqrt {d+e x}}{b}\right )+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 4 \left (\frac {2 (2 b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {2 \sqrt {a+b x} \sqrt {d+e x}}{b}\right )+\frac {6 d^2 \sqrt {a+b x}}{\sqrt {d+e x} (b d-a e)}\) |
(6*d^2*Sqrt[a + b*x])/((b*d - a*e)*Sqrt[d + e*x]) + 4*((2*Sqrt[a + b*x]*Sq rt[d + e*x])/b + (2*(2*b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b] *Sqrt[d + e*x])])/(b^(3/2)*Sqrt[e]))
3.9.46.3.1 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)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) ), x] + Simp[1/((m + 1)*(e*f - d*g)) Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a , b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] && !IntegerQ[n] && !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(88)=176\).
Time = 0.48 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.06
| method | result | size |
| default | \(-\frac {2 \sqrt {b x +a}\, \left (2 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{3} x -6 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d \,e^{2} x +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2} e x +2 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} d \,e^{2}-6 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,d^{2} e +4 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{3}-4 a \,e^{2} x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+4 b d e x \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-4 a d e \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+7 b \,d^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{b \sqrt {b e}\, \left (a e -b d \right ) \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {e x +d}}\) | \(438\) |
-2*(b*x+a)^(1/2)*(2*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+ a*e+b*d)/(b*e)^(1/2))*a^2*e^3*x-6*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2 )*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d*e^2*x+4*ln(1/2*(2*b*e*x+2*((b*x+ a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^2*e*x+2*ln(1/2*( 2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*d* e^2-6*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e) ^(1/2))*a*b*d^2*e+4*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+ a*e+b*d)/(b*e)^(1/2))*b^2*d^3-4*a*e^2*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2 )+4*b*d*e*x*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-4*a*d*e*((b*x+a)*(e*x+d))^ (1/2)*(b*e)^(1/2)+7*b*d^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/b/(b*e)^(1/ 2)/(a*e-b*d)/((b*x+a)*(e*x+d))^(1/2)/(e*x+d)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (88) = 176\).
Time = 0.39 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.29 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\left [-\frac {2 \, {\left ({\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}, -\frac {2 \, {\left (2 \, {\left (2 \, b^{2} d^{3} - 3 \, a b d^{2} e + a^{2} d e^{2} + {\left (2 \, b^{2} d^{2} e - 3 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - {\left (7 \, b^{2} d^{2} e - 4 \, a b d e^{2} + 4 \, {\left (b^{2} d e^{2} - a b e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}\right )}}{b^{3} d^{2} e - a b^{2} d e^{2} + {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x}\right ] \]
[-2*((2*b^2*d^3 - 3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*a*b*d*e^2 + a ^2*e^3)*x)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4 *(2*b*e*x + b*d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - (7*b^2*d^2*e - 4*a*b*d*e^2 + 4*(b^2*d*e^2 - a*b*e^3)*x)*sq rt(b*x + a)*sqrt(e*x + d))/(b^3*d^2*e - a*b^2*d*e^2 + (b^3*d*e^2 - a*b^2*e ^3)*x), -2*(2*(2*b^2*d^3 - 3*a*b*d^2*e + a^2*d*e^2 + (2*b^2*d^2*e - 3*a*b* d*e^2 + a^2*e^3)*x)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e) *sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)* x)) - (7*b^2*d^2*e - 4*a*b*d*e^2 + 4*(b^2*d*e^2 - a*b*e^3)*x)*sqrt(b*x + a )*sqrt(e*x + d))/(b^3*d^2*e - a*b^2*d*e^2 + (b^3*d*e^2 - a*b^2*e^3)*x)]
\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.82 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {4 \, {\left (b^{3} d e^{3} - a b^{2} e^{4}\right )} {\left (b x + a\right )}}{b^{3} d e^{2} {\left | b \right |} - a b^{2} e^{3} {\left | b \right |}} + \frac {7 \, b^{4} d^{2} e^{2} - 8 \, a b^{3} d e^{3} + 4 \, a^{2} b^{2} e^{4}}{b^{3} d e^{2} {\left | b \right |} - a b^{2} e^{3} {\left | b \right |}}\right )}}{\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e}} - \frac {8 \, {\left (2 \, b d - a e\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} {\left | b \right |}} \]
2*sqrt(b*x + a)*(4*(b^3*d*e^3 - a*b^2*e^4)*(b*x + a)/(b^3*d*e^2*abs(b) - a *b^2*e^3*abs(b)) + (7*b^4*d^2*e^2 - 8*a*b^3*d*e^3 + 4*a^2*b^2*e^4)/(b^3*d* e^2*abs(b) - a*b^2*e^3*abs(b)))/sqrt(b^2*d + (b*x + a)*b*e - a*b*e) - 8*(2 *b*d - a*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/(sqrt(b*e)*abs(b))
Timed out. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} (d+e x)^{3/2}} \, dx=\int \frac {15\,d^2+20\,d\,e\,x+8\,e^2\,x^2}{\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2}} \,d x \]