\(\int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx\) [342]

Optimal result

Integrand size = 19, antiderivative size = 183 \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=-\frac {35 (b c-a d)^3 \sqrt {a+b x} \sqrt {c+d x}}{64 d^4}+\frac {35 (b c-a d)^2 (a+b x)^{3/2} \sqrt {c+d x}}{96 d^3}-\frac {7 (b c-a d) (a+b x)^{5/2} \sqrt {c+d x}}{24 d^2}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}+\frac {35 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 \sqrt {b} d^{9/2}} \] Output:

-35/64*(-a*d+b*c)^3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/d^4+35/96*(-a*d+b*c)^2*(b* 
x+a)^(3/2)*(d*x+c)^(1/2)/d^3-7/24*(-a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/d 
^2+1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/d+35/64*(-a*d+b*c)^4*arctanh(d^(1/2)*(b 
*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (279 a^3 d^3+a^2 b d^2 (-511 c+326 d x)+a b^2 d \left (385 c^2-252 c d x+200 d^2 x^2\right )+b^3 \left (-105 c^3+70 c^2 d x-56 c d^2 x^2+48 d^3 x^3\right )\right )}{192 d^4}+\frac {35 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{64 \sqrt {b} d^{9/2}} \] Input:

Integrate[(a + b*x)^(7/2)/Sqrt[c + d*x],x]
 

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(279*a^3*d^3 + a^2*b*d^2*(-511*c + 326*d*x) + 
 a*b^2*d*(385*c^2 - 252*c*d*x + 200*d^2*x^2) + b^3*(-105*c^3 + 70*c^2*d*x 
- 56*c*d^2*x^2 + 48*d^3*x^3)))/(192*d^4) + (35*(b*c - a*d)^4*ArcTanh[(Sqrt 
[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(64*Sqrt[b]*d^(9/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {60, 60, 60, 60, 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.

\(\displaystyle \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 d}-\frac {7 (b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 d}\)

Input:

Int[(a + b*x)^(7/2)/Sqrt[c + d*x],x]
 

Output:

((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*d) - (7*(b*c - a*d)*(((a + b*x)^(5/2)*S 
qrt[c + d*x])/(3*d) - (5*(b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*d 
) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh 
[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4* 
d)))/(6*d)))/(8*d)
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

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]
 

rule 221

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]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {\left (b x +a \right )^{\frac {7}{2}} \sqrt {x d +c}}{4 d}-\frac {7 \left (-a d +b c \right ) \left (\frac {\left (b x +a \right )^{\frac {5}{2}} \sqrt {x d +c}}{3 d}-\frac {5 \left (-a d +b c \right ) \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \sqrt {x d +c}}{2 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {x d +c}}{d}-\frac {\left (-a d +b c \right ) \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {d b}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 d \sqrt {b x +a}\, \sqrt {x d +c}\, \sqrt {d b}}\right )}{4 d}\right )}{6 d}\right )}{8 d}\)	\(206\)

Input:

int((b*x+a)^(7/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*(b*x+a)^(7/2)*(d*x+c)^(1/2)/d-7/8*(-a*d+b*c)/d*(1/3*(b*x+a)^(5/2)*(d*x 
+c)^(1/2)/d-5/6*(-a*d+b*c)/d*(1/2*(b*x+a)^(3/2)*(d*x+c)^(1/2)/d-3/4*(-a*d+ 
b*c)/d*((b*x+a)^(1/2)*(d*x+c)^(1/2)/d-1/2*(-a*d+b*c)/d*((b*x+a)*(d*x+c))^( 
1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)*ln((1/2*a*d+1/2*b*c+b*d*x)/(d*b)^(1/2)+(b 
*d*x^2+(a*d+b*c)*x+a*c)^(1/2))/(d*b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.96 \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\left [\frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, b d^{5}}, -\frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{3} - 105 \, b^{4} c^{3} d + 385 \, a b^{3} c^{2} d^{2} - 511 \, a^{2} b^{2} c d^{3} + 279 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 25 \, a b^{3} d^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 126 \, a b^{3} c d^{3} + 163 \, a^{2} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, b d^{5}}\right ] \] Input:

integrate((b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

[1/768*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 
 a^4*d^4)*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*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511 
*a^2*b^2*c*d^3 + 279*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(3 
5*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d 
*x + c))/(b*d^5), -1/384*(105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 
 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sq 
rt(-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*(48*b^4*d^4*x^3 - 105*b^4*c^3*d + 385*a*b^3*c^2*d^2 - 511* 
a^2*b^2*c*d^3 + 279*a^3*b*d^4 - 8*(7*b^4*c*d^3 - 25*a*b^3*d^4)*x^2 + 2*(35 
*b^4*c^2*d^2 - 126*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d* 
x + c))/(b*d^5)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\int \frac {\left (a + b x\right )^{\frac {7}{2}}}{\sqrt {c + d x}}\, dx \] Input:

integrate((b*x+a)**(7/2)/(d*x+c)**(1/2),x)
 

Output:

Integral((a + b*x)**(7/2)/sqrt(c + d*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

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
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b d} - \frac {7 \, {\left (b c d^{5} - a d^{6}\right )}}{b d^{7}}\right )} + \frac {35 \, {\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )}}{b d^{7}}\right )} - \frac {105 \, {\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )}}{b d^{7}}\right )} \sqrt {b x + a} - \frac {105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} d^{4}}\right )} b}{192 \, {\left | b \right |}} \] Input:

integrate((b*x+a)^(7/2)/(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

1/192*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b 
*x + a)/(b*d) - 7*(b*c*d^5 - a*d^6)/(b*d^7)) + 35*(b^2*c^2*d^4 - 2*a*b*c*d 
^5 + a^2*d^6)/(b*d^7)) - 105*(b^3*c^3*d^3 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^ 
5 - a^3*d^6)/(b*d^7))*sqrt(b*x + a) - 105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2 
*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*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^4))*b/abs(b)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\int \frac {{\left (a+b\,x\right )}^{7/2}}{\sqrt {c+d\,x}} \,d x \] Input:

int((a + b*x)^(7/2)/(c + d*x)^(1/2),x)
 

Output:

int((a + b*x)^(7/2)/(c + d*x)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 471, normalized size of antiderivative = 2.57 \[ \int \frac {(a+b x)^{7/2}}{\sqrt {c+d x}} \, dx=\frac {279 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{3} b \,d^{4}-511 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}+326 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x +385 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}-252 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x +200 \sqrt {d x +c}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}-105 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{3} d +70 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x -56 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}+105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{4} d^{4}-420 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{3} b c \,d^{3}+630 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} b^{2} c^{2} d^{2}-420 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a \,b^{3} c^{3} d +105 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{4} c^{4}}{192 b \,d^{5}} \] Input:

int((b*x+a)^(7/2)/(d*x+c)^(1/2),x)
 

Output:

(279*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*d**4 - 511*sqrt(c + d*x)*sqrt(a + 
b*x)*a**2*b**2*c*d**3 + 326*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*d**4*x + 
 385*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*c**2*d**2 - 252*sqrt(c + d*x)*sqrt 
(a + b*x)*a*b**3*c*d**3*x + 200*sqrt(c + d*x)*sqrt(a + b*x)*a*b**3*d**4*x* 
*2 - 105*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c**3*d + 70*sqrt(c + d*x)*sqrt(a 
 + b*x)*b**4*c**2*d**2*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*b**4*c*d**3*x**2 
 + 48*sqrt(c + d*x)*sqrt(a + b*x)*b**4*d**4*x**3 + 105*sqrt(d)*sqrt(b)*log 
((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*d** 
4 - 420*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 
)/sqrt(a*d - b*c))*a**3*b*c*d**3 + 630*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a 
 + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b**2*c**2*d**2 - 42 
0*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt 
(a*d - b*c))*a*b**3*c**3*d + 105*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x 
) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**4*c**4)/(192*b*d**5)