Integrand size = 38, antiderivative size = 122 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 (7 b d-5 a e) \sqrt {a+b x} \sqrt {d+e x}}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}+\frac {2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{5/2} \sqrt {e}} \]
2*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e *x+d)^(1/2))/b^(5/2)/e^(1/2)+4*e*(b*x+a)^(3/2)*(e*x+d)^(1/2)/b^2+2*(-5*a*e +7*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^2
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-3 a e+2 b e x)}{b^2}+\frac {2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{b^{5/2} \sqrt {e}} \]
(2*Sqrt[a + b*x]*Sqrt[d + e*x]*(7*b*d - 3*a*e + 2*b*e*x))/b^2 + (2*(8*b^2* d^2 - 8*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt [a + b*x])])/(b^(5/2)*Sqrt[e])
Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1194, 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.
| \(\displaystyle \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1194 |
| \(\displaystyle \frac {\int \frac {2 e \left (15 b^2 d^2-6 a b e d-2 a^2 e^2+2 b e (7 b d-5 a e) x\right )}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b^2 e}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {15 b^2 d^2-6 a b e d-2 a^2 e^2+2 b e (7 b d-5 a e) x}{\sqrt {a+b x} \sqrt {d+e x}}dx}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx+2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-5 a e)}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}+2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-5 a e)}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{\sqrt {b} \sqrt {e}}+2 \sqrt {a+b x} \sqrt {d+e x} (7 b d-5 a e)}{b^2}+\frac {4 e (a+b x)^{3/2} \sqrt {d+e x}}{b^2}\) |
(4*e*(a + b*x)^(3/2)*Sqrt[d + e*x])/b^2 + (2*(7*b*d - 5*a*e)*Sqrt[a + b*x] *Sqrt[d + e*x] + (2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*ArcTanh[(Sqrt[e]*S qrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(Sqrt[b]*Sqrt[e]))/b^2
3.9.45.3.1 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] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x )^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 *p + 1)) Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) *(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ [p, 0] && !IntegerQ[m] && !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(102)=204\).
Time = 0.48 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.02
| method | result | size |
| default | \(\frac {\left (4 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b e x +3 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{2}-8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d e +8 \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2}-6 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a e +14 \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b d \right ) \sqrt {e x +d}\, \sqrt {b x +a}}{\sqrt {b e}\, b^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}}\) | \(247\) |
(4*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b*e*x+3*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^2-8*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+8*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^2-6*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*a*e+14*(b*e)^(1/2)*((b*x+a)*( e*x+d))^(1/2)*b*d)*(e*x+d)^(1/2)*(b*x+a)^(1/2)/(b*e)^(1/2)/b^2/((b*x+a)*(e *x+d))^(1/2)
Time = 0.41 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.52 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\left [\frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{2 \, b^{3} e}, -\frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) - 2 \, {\left (2 \, b^{2} e^{2} x + 7 \, b^{2} d e - 3 \, a b e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{b^{3} e}\right ] \]
[1/2*((8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)*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) + 4*(2*b^2*e^2*x + 7*b^2*d*e - 3*a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3*e), -((8*b^2*d^2 - 8*a*b*d* e + 3*a^2*e^2)*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)) - 2*(2*b^2*e^2*x + 7*b^2*d*e - 3*a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d))/(b^3 *e)]
\[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\int \frac {15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt {a + b x} \sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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*e-b*d>0)', see `assume?` for m ore detail
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.20 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {2 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} e}{b^{3}} + \frac {7 \, b^{6} d e^{2} - 5 \, a b^{5} e^{3}}{b^{8} e^{2}}\right )} - \frac {{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} b^{2}}\right )} b}{{\left | b \right |}} \]
2*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e/b^3 + (7*b^6*d*e^2 - 5*a*b^5*e^3)/(b^8*e^2)) - (8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^ 2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)) )/(sqrt(b*e)*b^2))*b/abs(b)
Time = 32.86 (sec) , antiderivative size = 893, normalized size of antiderivative = 7.32 \[ \int \frac {15 d^2+20 d e x+8 e^2 x^2}{\sqrt {a+b x} \sqrt {d+e x}} \, dx=\frac {\frac {\left (40\,b\,d^2+40\,a\,e\,d\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{e^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {160\,\sqrt {a}\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {\left (40\,b\,d^2+40\,a\,e\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {b^2}{e^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}-\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (12\,a^2\,b\,e^2+8\,a\,b^2\,d\,e+12\,b^3\,d^2\right )}{e^4\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (44\,a^2\,e^2+200\,a\,b\,d\,e+44\,b^2\,d^2\right )}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (12\,a^2\,e^2+8\,a\,b\,d\,e+12\,b^2\,d^2\right )}{b^2\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (44\,a^2\,e^2+200\,a\,b\,d\,e+44\,b^2\,d^2\right )}{b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}+\frac {\sqrt {a}\,\sqrt {d}\,\left (256\,a\,e+256\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {b^4}{e^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}}-\frac {60\,d^2\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{\sqrt {-b\,e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}\right )}{\sqrt {-b\,e}}-\frac {2\,\ln \left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {d+e\,x}-\sqrt {d}}-\sqrt {b}\right )\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+3\,b^2\,d^2\right )}{b^{5/2}\,\sqrt {e}}+\frac {\ln \left (\sqrt {b}+\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {d+e\,x}-\sqrt {d}}\right )\,\left (6\,a^2\,e^2+4\,a\,b\,d\,e+6\,b^2\,d^2\right )}{b^{5/2}\,\sqrt {e}}-\frac {40\,d\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e+b\,d\right )}{b^{3/2}\,\sqrt {e}} \]
(((40*b*d^2 + 40*a*d*e)*((a + b*x)^(1/2) - a^(1/2)))/(e^2*((d + e*x)^(1/2) - d^(1/2))) - (160*a^(1/2)*d^(3/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(e*((d + e*x)^(1/2) - d^(1/2))^2) + ((40*b*d^2 + 40*a*d*e)*((a + b*x)^(1/2) - a^( 1/2))^3)/(b*e*((d + e*x)^(1/2) - d^(1/2))^3))/(((a + b*x)^(1/2) - a^(1/2)) ^4/((d + e*x)^(1/2) - d^(1/2))^4 + b^2/e^2 - (2*b*((a + b*x)^(1/2) - a^(1/ 2))^2)/(e*((d + e*x)^(1/2) - d^(1/2))^2)) - ((((a + b*x)^(1/2) - a^(1/2))* (12*b^3*d^2 + 12*a^2*b*e^2 + 8*a*b^2*d*e))/(e^4*((d + e*x)^(1/2) - d^(1/2) )) - (((a + b*x)^(1/2) - a^(1/2))^3*(44*a^2*e^2 + 44*b^2*d^2 + 200*a*b*d*e ))/(e^3*((d + e*x)^(1/2) - d^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^7*(1 2*a^2*e^2 + 12*b^2*d^2 + 8*a*b*d*e))/(b^2*e*((d + e*x)^(1/2) - d^(1/2))^7) - (((a + b*x)^(1/2) - a^(1/2))^5*(44*a^2*e^2 + 44*b^2*d^2 + 200*a*b*d*e)) /(b*e^2*((d + e*x)^(1/2) - d^(1/2))^5) + (a^(1/2)*d^(1/2)*(256*a*e + 256*b *d)*((a + b*x)^(1/2) - a^(1/2))^4)/(e^2*((d + e*x)^(1/2) - d^(1/2))^4))/(( (a + b*x)^(1/2) - a^(1/2))^8/((d + e*x)^(1/2) - d^(1/2))^8 + b^4/e^4 - (4* b^3*((a + b*x)^(1/2) - a^(1/2))^2)/(e^3*((d + e*x)^(1/2) - d^(1/2))^2) + ( 6*b^2*((a + b*x)^(1/2) - a^(1/2))^4)/(e^2*((d + e*x)^(1/2) - d^(1/2))^4) - (4*b*((a + b*x)^(1/2) - a^(1/2))^6)/(e*((d + e*x)^(1/2) - d^(1/2))^6)) - (60*d^2*atan((b*((d + e*x)^(1/2) - d^(1/2)))/((-b*e)^(1/2)*((a + b*x)^(1/2 ) - a^(1/2)))))/(-b*e)^(1/2) - (2*log((e^(1/2)*((a + b*x)^(1/2) - a^(1/2)) )/((d + e*x)^(1/2) - d^(1/2)) - b^(1/2))*(3*a^2*e^2 + 3*b^2*d^2 + 2*a*b...