Integrand size = 24, antiderivative size = 302 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\frac {35 e (b d-a e) (b B d+2 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{8 b^5}+\frac {35 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{3/2}}{12 b^4}+\frac {7 e (b B d+2 A b e-3 a B e) \sqrt {a+b x} (d+e x)^{5/2}}{3 b^3 (b d-a e)}-\frac {2 (b B d+2 A b e-3 a B e) (d+e x)^{7/2}}{b^2 (b d-a e) \sqrt {a+b x}}-\frac {2 (A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{8 b^{11/2}} \]
-2/3*(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)/(b*x+a)^(3/2)+35/8*(-a*e+b*d)^2* (2*A*b*e-3*B*a*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(e*x+d)^(1/2 ))*e^(1/2)/b^(11/2)-2*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*d) /(b*x+a)^(1/2)+35/12*e*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(3/2)*(b*x+a)^(1/2) /b^4+7/3*e*(2*A*b*e-3*B*a*e+B*b*d)*(e*x+d)^(5/2)*(b*x+a)^(1/2)/b^3/(-a*e+b *d)+35/8*e*(-a*e+b*d)*(2*A*b*e-3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/ b^5
Time = 0.83 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (-2 A b \left (105 a^3 e^3+35 a^2 b e^2 (-5 d+4 e x)+7 a b^2 e \left (8 d^2-34 d e x+3 e^2 x^2\right )+b^3 \left (8 d^3+80 d^2 e x-39 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (315 a^4 e^3+210 a^3 b e^2 (-3 d+2 e x)+7 a^2 b^2 e \left (49 d^2-122 d e x+9 e^2 x^2\right )+b^4 x \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )-2 a b^3 \left (16 d^3-239 d^2 e x+69 d e^2 x^2+9 e^3 x^3\right )\right )\right )}{24 b^5 (a+b x)^{3/2}}+\frac {35 \sqrt {e} (b d-a e)^2 (b B d+2 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \sqrt {a+b x}}\right )}{8 b^{11/2}} \]
(Sqrt[d + e*x]*(-2*A*b*(105*a^3*e^3 + 35*a^2*b*e^2*(-5*d + 4*e*x) + 7*a*b^ 2*e*(8*d^2 - 34*d*e*x + 3*e^2*x^2) + b^3*(8*d^3 + 80*d^2*e*x - 39*d*e^2*x^ 2 - 6*e^3*x^3)) + B*(315*a^4*e^3 + 210*a^3*b*e^2*(-3*d + 2*e*x) + 7*a^2*b^ 2*e*(49*d^2 - 122*d*e*x + 9*e^2*x^2) + b^4*x*(-48*d^3 + 87*d^2*e*x + 38*d* e^2*x^2 + 8*e^3*x^3) - 2*a*b^3*(16*d^3 - 239*d^2*e*x + 69*d*e^2*x^2 + 9*e^ 3*x^3))))/(24*b^5*(a + b*x)^(3/2)) + (35*Sqrt[e]*(b*d - a*e)^2*(b*B*d + 2* A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a + b*x])]) /(8*b^(11/2))
Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.85, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {87, 57, 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.
\(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \int \frac {(d+e x)^{7/2}}{(a+b x)^{3/2}}dx}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{\sqrt {a+b x}}dx}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \int \frac {(d+e x)^{3/2}}{\sqrt {a+b x}}dx}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(-3 a B e+2 A b e+b B d) \left (\frac {7 e \left (\frac {5 (b d-a e) \left (\frac {3 (b d-a e) \left (\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (d+e x)^{3/2}}{2 b}\right )}{6 b}+\frac {\sqrt {a+b x} (d+e x)^{5/2}}{3 b}\right )}{b}-\frac {2 (d+e x)^{7/2}}{b \sqrt {a+b x}}\right )}{b (b d-a e)}-\frac {2 (d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^{3/2} (b d-a e)}\) |
(-2*(A*b - a*B)*(d + e*x)^(9/2))/(3*b*(b*d - a*e)*(a + b*x)^(3/2)) + ((b*B *d + 2*A*b*e - 3*a*B*e)*((-2*(d + e*x)^(7/2))/(b*Sqrt[a + b*x]) + (7*e*((S qrt[a + b*x]*(d + e*x)^(5/2))/(3*b) + (5*(b*d - a*e)*((Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b) + (3*(b*d - a*e)*((Sqrt[a + b*x]*Sqrt[d + e*x])/b + ((b* d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(b^(3/2 )*Sqrt[e])))/(4*b)))/(6*b)))/b))/(b*(b*d - a*e))
3.23.53.3.1 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] :> 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]
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)^(-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. \(1881\) vs. \(2(260)=520\).
Time = 1.10 (sec) , antiderivative size = 1882, normalized size of antiderivative = 6.23
1/48*(e*x+d)^(1/2)*(-315*B*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^5*e^4+956*B*a*b^3*d^2*e*x*(b*e)^(1/2)*((b*x +a)*(e*x+d))^(1/2)-560*A*a^2*b^2*e^3*x*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2) -320*A*b^4*d^2*e*x*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+840*B*a^3*b*e^3*x*( b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+700*A*a^2*b^2*d*e^2*(b*e)^(1/2)*((b*x+a )*(e*x+d))^(1/2)-224*A*a*b^3*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-126 0*B*a^3*b*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-315*B*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^3*b^2*e^4*x^ 2+105*B*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^5*d^3*e*x^2-420*A*a^3*b*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2 )-64*B*a*b^3*d^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+210*A*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*b^3*e^4*x ^2+210*A*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^5*d^2*e^2*x^2+16*B*b^4*e^3*x^4*(b*e)^(1/2)*((b*x+a)*(e*x+d))^ (1/2)+420*A*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^3*b^2*e^4*x-630*B*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^4*b*e^4*x+686*B*a^2*b^2*d^2*e*(b*e)^ (1/2)*((b*x+a)*(e*x+d))^(1/2)-420*A*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^4*d*e^3*x^2-1050*B*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*b^3*...
Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (260) = 520\).
Time = 1.42 (sec) , antiderivative size = 1271, normalized size of antiderivative = 4.21 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
[-1/96*(105*(B*a^2*b^3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4* b - 4*A*a^3*b^2)*d*e^2 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b ^4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A *a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7 *B*a^3*b^2 - 4*A*a^2*b^3)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*x)*sqrt(e /b)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b^2*e*x + b^2 *d + a*b*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(e/b) + 8*(b^2*d*e + a*b*e^2)* x) - 4*(8*B*b^4*e^3*x^4 - 16*(2*B*a*b^3 + A*b^4)*d^3 + 7*(49*B*a^2*b^2 - 1 6*A*a*b^3)*d^2*e - 70*(9*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(3*B*a^4 - 2*A *a^3*b)*e^3 + 2*(19*B*b^4*d*e^2 - 3*(3*B*a*b^3 - 2*A*b^4)*e^3)*x^3 + 3*(29 *B*b^4*d^2*e - 2*(23*B*a*b^3 - 13*A*b^4)*d*e^2 + 7*(3*B*a^2*b^2 - 2*A*a*b^ 3)*e^3)*x^2 - 2*(24*B*b^4*d^3 - (239*B*a*b^3 - 80*A*b^4)*d^2*e + 7*(61*B*a ^2*b^2 - 34*A*a*b^3)*d*e^2 - 70*(3*B*a^3*b - 2*A*a^2*b^2)*e^3)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/48*(105*(B*a^2*b^ 3*d^3 - (5*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e + (7*B*a^4*b - 4*A*a^3*b^2)*d*e^ 2 - (3*B*a^5 - 2*A*a^4*b)*e^3 + (B*b^5*d^3 - (5*B*a*b^4 - 2*A*b^5)*d^2*e + (7*B*a^2*b^3 - 4*A*a*b^4)*d*e^2 - (3*B*a^3*b^2 - 2*A*a^2*b^3)*e^3)*x^2 + 2*(B*a*b^4*d^3 - (5*B*a^2*b^3 - 2*A*a*b^4)*d^2*e + (7*B*a^3*b^2 - 4*A*a^2* b^3)*d*e^2 - (3*B*a^4*b - 2*A*a^3*b^2)*e^3)*x)*sqrt(-e/b)*arctan(1/2*(2*b* e*x + b*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(-e/b)/(b*e^2*x^2 + a*...
\[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {7}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1814 vs. \(2 (260) = 520\).
Time = 0.87 (sec) , antiderivative size = 1814, normalized size of antiderivative = 6.01 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
1/24*sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b* x + a)*B*e^3*abs(b)/b^7 + (19*B*b^21*d*e^6*abs(b) - 25*B*a*b^20*e^7*abs(b) + 6*A*b^21*e^7*abs(b))/(b^27*e^4)) + 3*(29*B*b^22*d^2*e^5*abs(b) - 84*B*a *b^21*d*e^6*abs(b) + 26*A*b^22*d*e^6*abs(b) + 55*B*a^2*b^20*e^7*abs(b) - 2 6*A*a*b^21*e^7*abs(b))/(b^27*e^4)) - 35/16*(sqrt(b*e)*B*b^3*d^3*abs(b) - 5 *sqrt(b*e)*B*a*b^2*d^2*e*abs(b) + 2*sqrt(b*e)*A*b^3*d^2*e*abs(b) + 7*sqrt( b*e)*B*a^2*b*d*e^2*abs(b) - 4*sqrt(b*e)*A*a*b^2*d*e^2*abs(b) - 3*sqrt(b*e) *B*a^3*e^3*abs(b) + 2*sqrt(b*e)*A*a^2*b*e^3*abs(b))*log((sqrt(b*e)*sqrt(b* x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2)/b^7 - 4/3*(3*sqrt(b*e)*B* b^8*d^6*abs(b) - 28*sqrt(b*e)*B*a*b^7*d^5*e*abs(b) + 10*sqrt(b*e)*A*b^8*d^ 5*e*abs(b) + 95*sqrt(b*e)*B*a^2*b^6*d^4*e^2*abs(b) - 50*sqrt(b*e)*A*a*b^7* d^4*e^2*abs(b) - 160*sqrt(b*e)*B*a^3*b^5*d^3*e^3*abs(b) + 100*sqrt(b*e)*A* a^2*b^6*d^3*e^3*abs(b) + 145*sqrt(b*e)*B*a^4*b^4*d^2*e^4*abs(b) - 100*sqrt (b*e)*A*a^3*b^5*d^2*e^4*abs(b) - 68*sqrt(b*e)*B*a^5*b^3*d*e^5*abs(b) + 50* sqrt(b*e)*A*a^4*b^4*d*e^5*abs(b) + 13*sqrt(b*e)*B*a^6*b^2*e^6*abs(b) - 10* sqrt(b*e)*A*a^5*b^3*e^6*abs(b) - 6*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sq rt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*b^6*d^5*abs(b) + 48*sqrt(b*e)*(sqrt (b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e))^2*B*a*b^5*d^4*e *abs(b) - 18*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x + a) - sqrt(b^2*d + (b*x + a)*b *e - a*b*e))^2*A*b^6*d^4*e*abs(b) - 132*sqrt(b*e)*(sqrt(b*e)*sqrt(b*x +...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]