Integrand size = 19, antiderivative size = 806 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}-\frac {9 \left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt {a+b x} \sqrt [6]{c+d x}}{d \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}-\frac {9 \sqrt [4]{3} \sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}-\frac {3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 d^2 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]
-6*(b*x+a)^(1/2)/d/(d*x+c)^(1/6)-9*b^(1/3)*(d*x+c)^(1/6)*(1+3^(1/2))*(b*x+ a)^(1/2)/d/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))-9*3^(1/4)* b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/6)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^( 1/3))*(((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1-3^(1/2)))^2/((-a*d+b*c)^ (1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2)/((-a*d+b*c)^(1/3)-b^(1/3 )*(d*x+c)^(1/3)*(1-3^(1/2)))*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^ (1/2)))*EllipticE((1-((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1-3^(1/2)))^ 2/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2 )+1/4*2^(1/2))*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b ^(2/3)*(d*x+c)^(2/3))/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2))) ^2)^(1/2)/d^2/(b*x+a)^(1/2)/(-b^(1/3)*(d*x+c)^(1/3)*((-a*d+b*c)^(1/3)-b^(1 /3)*(d*x+c)^(1/3))/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2) ^(1/2)-3/2*3^(3/4)*b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/6)*((-a*d+b*c)^(1/3 )-b^(1/3)*(d*x+c)^(1/3))*(((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1-3^(1/ 2)))^2/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2)/((-a* d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1-3^(1/2)))*((-a*d+b*c)^(1/3)-b^(1/3)* (d*x+c)^(1/3)*(1+3^(1/2)))*EllipticF((1-((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^ (1/3)*(1-3^(1/2)))^2/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^ 2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*(1-3^(1/2))*(((-a*d+b*c)^(2/3)+b^(1/3)*( -a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b^(2/3)*(d*x+c)^(2/3))/((-a*d+b*c)^(1/3)-...
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\frac {2 (a+b x)^{3/2} \left (\frac {b (c+d x)}{b c-a d}\right )^{7/6} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {3}{2},\frac {5}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (c+d x)^{7/6}} \]
(2*(a + b*x)^(3/2)*((b*(c + d*x))/(b*c - a*d))^(7/6)*Hypergeometric2F1[7/6 , 3/2, 5/2, (d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(c + d*x)^(7/6))
Time = 0.72 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {57, 73, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle \frac {3 b \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}}dx}{d}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {18 b \int \frac {(c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{d^2}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {18 b \left (-\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\int -\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}\right )}{d^2}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {18 b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}\right )}{d^2}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {18 b \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {18 b \left (\frac {-\frac {\left (1+\sqrt {3}\right ) \sqrt [6]{c+d x} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}} d}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}-\frac {\sqrt [4]{3} \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{d^2}-\frac {6 \sqrt {a+b x}}{d \sqrt [6]{c+d x}}\) |
(-6*Sqrt[a + b*x])/(d*(c + d*x)^(1/6)) + (18*b*((-(((1 + Sqrt[3])*d*(c + d *x)^(1/6)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/((b*c - a*d)^(1/3) - (1 + S qrt[3])*b^(1/3)*(c + d*x)^(1/3))) - (3^(1/4)*(b*c - a*d)^(1/3)*(c + d*x)^( 1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/(( b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[Arc Cos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a* d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(Sqr t[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)) )/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]))/(2*b^(2/3)) - ((1 - Sqrt[3])*(b*c - a*d)^(1/ 3)*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b* c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2] *EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/ 3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqr t[3])/4])/(4*3^(1/4)*b^(2/3)*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^ (1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/ 3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/d^2
3.18.62.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
\[\int \frac {\sqrt {b x +a}}{\left (d x +c \right )^{\frac {7}{6}}}d x\]
\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\int \frac {\sqrt {a + b x}}{\left (c + d x\right )^{\frac {7}{6}}}\, dx \]
\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\int { \frac {\sqrt {b x + a}}{{\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\int \frac {\sqrt {a+b\,x}}{{\left (c+d\,x\right )}^{7/6}} \,d x \]
\[ \int \frac {\sqrt {a+b x}}{(c+d x)^{7/6}} \, dx=\frac {-2 \sqrt {d x +c}\, \sqrt {b x +a}\, b c -2 \sqrt {d x +c}\, \sqrt {b x +a}\, b d x +\left (d x +c \right )^{\frac {2}{3}} \left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \sqrt {b x +a}}{b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}d x \right ) a^{2} d^{2}-2 \left (d x +c \right )^{\frac {2}{3}} \left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \sqrt {b x +a}}{b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}d x \right ) a b c d +\left (d x +c \right )^{\frac {2}{3}} \left (\int \frac {\left (d x +c \right )^{\frac {5}{6}} \sqrt {b x +a}}{b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}d x \right ) b^{2} c^{2}}{\left (d x +c \right )^{\frac {2}{3}} d \left (a d -b c \right )} \]
( - 2*sqrt(c + d*x)*sqrt(a + b*x)*b*c - 2*sqrt(c + d*x)*sqrt(a + b*x)*b*d* x + (c + d*x)**(2/3)*int(((c + d*x)**(5/6)*sqrt(a + b*x))/(a*c**2 + 2*a*c* d*x + a*d**2*x**2 + b*c**2*x + 2*b*c*d*x**2 + b*d**2*x**3),x)*a**2*d**2 - 2*(c + d*x)**(2/3)*int(((c + d*x)**(5/6)*sqrt(a + b*x))/(a*c**2 + 2*a*c*d* x + a*d**2*x**2 + b*c**2*x + 2*b*c*d*x**2 + b*d**2*x**3),x)*a*b*c*d + (c + d*x)**(2/3)*int(((c + d*x)**(5/6)*sqrt(a + b*x))/(a*c**2 + 2*a*c*d*x + a* d**2*x**2 + b*c**2*x + 2*b*c*d*x**2 + b*d**2*x**3),x)*b**2*c**2)/((c + d*x )**(2/3)*d*(a*d - b*c))