Integrand size = 20, antiderivative size = 315 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \] Output:
-1/512*(-a*d+b*c)^4*(5*a*d+7*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^4+1/76 8*(-a*d+b*c)^3*(5*a*d+7*b*c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^3-1/960*(-a *d+b*c)^2*(5*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d^2-1/160*(-a*d+b* c)*(5*a*d+7*b*c)*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^3/d-1/60*(5*a*d+7*b*c)*(b*x +a)^(7/2)*(d*x+c)^(3/2)/b^2/d+1/6*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b/d+1/512*(- a*d+b*c)^5*(5*a*d+7*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/ 2))/b^(7/2)/d^(9/2)
Time = 0.62 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.93 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (49 c+10 d x)+10 a^3 b^2 d^3 \left (15 c^2+16 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-91 c^3+58 c^2 d x+564 c d^2 x^2+360 d^3 x^3\right )+a b^4 d \left (415 c^4-272 c^3 d x+216 c^2 d^2 x^2+4448 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^4}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \] Input:
Integrate[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]
Output:
(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(49*c + 10*d*x) + 1 0*a^3*b^2*d^3*(15*c^2 + 16*c*d*x + 4*d^2*x^2) + 6*a^2*b^3*d^2*(-91*c^3 + 5 8*c^2*d*x + 564*c*d^2*x^2 + 360*d^3*x^3) + a*b^4*d*(415*c^4 - 272*c^3*d*x + 216*c^2*d^2*x^2 + 4448*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(-105*c^5 + 70*c^ 4*d*x - 56*c^3*d^2*x^2 + 48*c^2*d^3*x^3 + 1664*c*d^4*x^4 + 1280*d^5*x^5))) /(7680*b^3*d^4) + ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(9/2))
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {90, 60, 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.
\(\displaystyle \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
Input:
Int[x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x]
Output:
((a + b*x)^(7/2)*(c + d*x)^(5/2))/(6*b*d) - ((7*b*c + 5*a*d)*(((a + b*x)^( 7/2)*(c + d*x)^(3/2))/(5*b) + (3*(b*c - a*d)*(((a + b*x)^(7/2)*Sqrt[c + d* x])/(4*b) + ((b*c - a*d)*(((a + b*x)^(5/2)*Sqrt[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*b)))/(10*b)))/ (12*b*d)
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*(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]
Leaf count of result is larger than twice the leaf count of optimal. \(1036\) vs. \(2(265)=530\).
Time = 0.23 (sec) , antiderivative size = 1037, normalized size of antiderivative = 3.29
Input:
int(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-8896*a*b^4*c*d^4*x^3*((b*x+a)*(d*x+ c))^(1/2)*(d*b)^(1/2)-6768*a^2*b^3*c*d^4*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b) ^(1/2)-432*a*b^4*c^2*d^3*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-320*(d*b) ^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^2*c*d^4*x-696*(d*b)^(1/2)*((b*x+a)*(d *x+c))^(1/2)*a^2*b^3*c^2*d^3*x+544*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b ^4*c^3*d^2*x-270*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d +b*c)/(d*b)^(1/2))*a^5*b*c*d^5+225*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/ 2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^4*b^2*c^2*d^4+300*ln(1/2*(2*b*d*x+2 *((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b^3*c^3*d^3 -675*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^ (1/2))*a^2*b^4*c^4*d^2+450*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b) ^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^5*c^5*d-6400*a*b^4*d^5*x^4*((b*x+a)*(d*x+ c))^(1/2)*(d*b)^(1/2)-3328*b^5*c*d^4*x^4*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 2)-4320*a^2*b^3*d^5*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-96*b^5*c^2*d^3 *x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-2560*b^5*d^5*x^5*((b*x+a)*(d*x+c) )^(1/2)*(d*b)^(1/2)+75*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/ 2)+a*d+b*c)/(d*b)^(1/2))*a^6*d^6-105*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^( 1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*b^6*c^6-150*(d*b)^(1/2)*((b*x+a)*(d *x+c))^(1/2)*a^5*d^5+210*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^5*c^5-300*( d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^2*c^2*d^3+1092*(d*b)^(1/2)*((b...
Time = 0.13 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.84 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx =\text {Too large to display} \] Input:
integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="fricas")
Output:
[-1/30720*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^ 3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*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)*s qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(1280* b^6*d^6*x^5 - 105*b^6*c^5*d + 415*a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 15 0*a^3*b^3*c^2*d^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d^5 + 25*a*b^5*d^6)*x^4 + 16*(3*b^6*c^2*d^4 + 278*a*b^5*c*d^5 + 135*a^2*b^4*d^6 )*x^3 - 8*(7*b^6*c^3*d^3 - 27*a*b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^ 3*d^6)*x^2 + 2*(35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2*d^4 + 80*a^3*b^3*c*d^5 - 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d ^5), -1/15360*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^ 3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*sqrt(-b*d )*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-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*(1280*b^6*d^6*x^5 - 1 05*b^6*c^5*d + 415*a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 150*a^3*b^3*c^2*d ^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d^5 + 25*a*b^5*d^6)* x^4 + 16*(3*b^6*c^2*d^4 + 278*a*b^5*c*d^5 + 135*a^2*b^4*d^6)*x^3 - 8*(7*b^ 6*c^3*d^3 - 27*a*b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^3*d^6)*x^2 + 2* (35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2*d^4 + 80*a^3*b^3*c*d ^5 - 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^5)]
\[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x*(b*x+a)**(5/2)*(d*x+c)**(3/2),x)
Output:
Integral(x*(a + b*x)**(5/2)*(c + d*x)**(3/2), x)
Exception generated. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),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
Leaf count of result is larger than twice the leaf count of optimal. 2272 vs. \(2 (265) = 530\).
Time = 0.40 (sec) , antiderivative size = 2272, normalized size of antiderivative = 7.21 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x, algorithm="giac")
Output:
1/7680*(120*(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^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2 *d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqr t(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c *d^3 - 35*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)*b^2*d^3))*a*c*abs(b) + 120*(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^3 + (b^12*c*d^ 5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2*d^4 + 14*a*b^12*c*d^5 - 163*a ^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b ^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqrt(b*x + a) + 3*(5*b^4*c^4 + 4* a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 35*a^4*d^4)*log(abs(-sq rt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b ^2*d^3))*a^2*d*abs(b)/b + (sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(2*(b *x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + (b^30*c*d^9 - 61*a*b^29*d^10)/(b^ 34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30*c*d^9 - 417*a^2*b^29*d^10)/(b^34 *d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2*d^8 + 183*a^2*b^30*c*d^9 - 3481 *a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5*(21*b^33*c^4*d^6 + 56*a*b^32*c^ 3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^30*c*d^9 - 2279*a^4*b^29*d^10)/(b ^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 + 35*a*b^33*c^4*d^6 + 50*a^2...
Timed out. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:
int(x*(a + b*x)^(5/2)*(c + d*x)^(3/2),x)
Output:
int(x*(a + b*x)^(5/2)*(c + d*x)^(3/2), x)
Time = 0.25 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.73 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx =\text {Too large to display} \] Input:
int(x*(b*x+a)^(5/2)*(d*x+c)^(3/2),x)
Output:
(75*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*d**6 - 245*sqrt(c + d*x)*sqrt(a + b *x)*a**4*b**2*c*d**5 - 50*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*d**6*x + 1 50*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*c**2*d**4 + 160*sqrt(c + d*x)*sqr t(a + b*x)*a**3*b**3*c*d**5*x + 40*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*d **6*x**2 - 546*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c**3*d**3 + 348*sqrt( c + d*x)*sqrt(a + b*x)*a**2*b**4*c**2*d**4*x + 3384*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c*d**5*x**2 + 2160*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*d **6*x**3 + 415*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*c**4*d**2 - 272*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*c**3*d**3*x + 216*sqrt(c + d*x)*sqrt(a + b*x)*a *b**5*c**2*d**4*x**2 + 4448*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*c*d**5*x**3 + 3200*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*d**6*x**4 - 105*sqrt(c + d*x)*s qrt(a + b*x)*b**6*c**5*d + 70*sqrt(c + d*x)*sqrt(a + b*x)*b**6*c**4*d**2*x - 56*sqrt(c + d*x)*sqrt(a + b*x)*b**6*c**3*d**3*x**2 + 48*sqrt(c + d*x)*s qrt(a + b*x)*b**6*c**2*d**4*x**3 + 1664*sqrt(c + d*x)*sqrt(a + b*x)*b**6*c *d**5*x**4 + 1280*sqrt(c + d*x)*sqrt(a + b*x)*b**6*d**6*x**5 - 75*sqrt(d)* sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c ))*a**6*d**6 + 270*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sq rt(c + d*x))/sqrt(a*d - b*c))*a**5*b*c*d**5 - 225*sqrt(d)*sqrt(b)*log((sqr t(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**4*b**2*c** 2*d**4 - 300*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(...