\(\int (a+b x)^{5/2} (c+d x)^{5/2} \, dx\) [331]

Optimal result

Integrand size = 19, antiderivative size = 262 \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\frac {5 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^3}-\frac {5 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^2}+\frac {(b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}{192 b^3 d}+\frac {(b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}{32 b^3}+\frac {(b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}{12 b^2}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}-\frac {5 (b c-a d)^6 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{7/2}} \] Output:

5/512*(-a*d+b*c)^5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^3-5/768*(-a*d+b*c)^4* 
(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^2+1/192*(-a*d+b*c)^3*(b*x+a)^(5/2)*(d*x+ 
c)^(1/2)/b^3/d+1/32*(-a*d+b*c)^2*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^3+1/12*(-a* 
d+b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2+1/6*(b*x+a)^(7/2)*(d*x+c)^(5/2)/b-5 
/512*(-a*d+b*c)^6*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^( 
7/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.08 \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 a^5 d^5-5 a^4 b d^4 (17 c+2 d x)+2 a^3 b^2 d^3 \left (99 c^2+28 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (33 c^3+198 c^2 d x+212 c d^2 x^2+72 d^3 x^3\right )+a b^4 d \left (-85 c^4+56 c^3 d x+1272 c^2 d^2 x^2+1696 c d^3 x^3+640 d^4 x^4\right )+b^5 \left (15 c^5-10 c^4 d x+8 c^3 d^2 x^2+432 c^2 d^3 x^3+640 c d^4 x^4+256 d^5 x^5\right )\right )}{1536 b^3 d^3}-\frac {5 (b c-a d)^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{512 b^{7/2} d^{7/2}} \] Input:

Integrate[(a + b*x)^(5/2)*(c + d*x)^(5/2),x]
 

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*a^5*d^5 - 5*a^4*b*d^4*(17*c + 2*d*x) + 2* 
a^3*b^2*d^3*(99*c^2 + 28*c*d*x + 4*d^2*x^2) + 6*a^2*b^3*d^2*(33*c^3 + 198* 
c^2*d*x + 212*c*d^2*x^2 + 72*d^3*x^3) + a*b^4*d*(-85*c^4 + 56*c^3*d*x + 12 
72*c^2*d^2*x^2 + 1696*c*d^3*x^3 + 640*d^4*x^4) + b^5*(15*c^5 - 10*c^4*d*x 
+ 8*c^3*d^2*x^2 + 432*c^2*d^3*x^3 + 640*c*d^4*x^4 + 256*d^5*x^5)))/(1536*b 
^3*d^3) - (5*(b*c - a*d)^6*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a 
 + b*x])])/(512*b^(7/2)*d^(7/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {60, 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.

\(\displaystyle \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 (b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b}\)

Input:

Int[(a + b*x)^(5/2)*(c + d*x)^(5/2),x]
 

Output:

((a + b*x)^(7/2)*(c + d*x)^(5/2))/(6*b) + (5*(b*c - 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)
 

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 = 0.15 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.04

method	result	size
default	\(\frac {\left (b x +a \right )^{\frac {5}{2}} \left (x d +c \right )^{\frac {7}{2}}}{6 d}-\frac {5 \left (-a d +b c \right ) \left (\frac {\left (b x +a \right )^{\frac {3}{2}} \left (x d +c \right )^{\frac {7}{2}}}{5 d}-\frac {3 \left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (x d +c \right )^{\frac {7}{2}}}{4 d}-\frac {\left (-a d +b c \right ) \left (\frac {\sqrt {b x +a}\, \left (x d +c \right )^{\frac {5}{2}}}{3 b}-\frac {5 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \left (x d +c \right )^{\frac {3}{2}}}{2 b}-\frac {3 \left (a d -b c \right ) \left (\frac {\sqrt {b x +a}\, \sqrt {x d +c}}{b}-\frac {\left (a d -b c \right ) \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d x}{\sqrt {d b}}+\sqrt {b d \,x^{2}+\left (a d +b c \right ) x +a c}\right )}{2 b \sqrt {x d +c}\, \sqrt {b x +a}\, \sqrt {d b}}\right )}{4 b}\right )}{6 b}\right )}{8 d}\right )}{10 d}\right )}{12 d}\)	\(272\)

Input:

int((b*x+a)^(5/2)*(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/6/d*(b*x+a)^(5/2)*(d*x+c)^(7/2)-5/12*(-a*d+b*c)/d*(1/5/d*(b*x+a)^(3/2)*( 
d*x+c)^(7/2)-3/10*(-a*d+b*c)/d*(1/4/d*(b*x+a)^(1/2)*(d*x+c)^(7/2)-1/8*(-a* 
d+b*c)/d*(1/3*(b*x+a)^(1/2)*(d*x+c)^(5/2)/b-5/6*(a*d-b*c)/b*(1/2*(b*x+a)^( 
1/2)*(d*x+c)^(3/2)/b-3/4*(a*d-b*c)/b*((b*x+a)^(1/2)*(d*x+c)^(1/2)/b-1/2*(a 
*d-b*c)/b*((b*x+a)*(d*x+c))^(1/2)/(d*x+c)^(1/2)/(b*x+a)^(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (212) = 424\).

Time = 0.14 (sec) , antiderivative size = 882, normalized size of antiderivative = 3.37 \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/6144*(15*(b^6*c^6 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^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)*sqrt(b*d) 
*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(256*b^6*d^6*x 
^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d^2 + 198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c 
^2*d^4 - 85*a^4*b^2*c*d^5 + 15*a^5*b*d^6 + 640*(b^6*c*d^5 + a*b^5*d^6)*x^4 
 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 + 27*a^2*b^4*d^6)*x^3 + 8*(b^6*c^3 
*d^3 + 159*a*b^5*c^2*d^4 + 159*a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6 
*c^4*d^2 - 28*a*b^5*c^3*d^3 - 594*a^2*b^4*c^2*d^4 - 28*a^3*b^3*c*d^5 + 5*a 
^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^4), 1/3072*(15*(b^6*c^6 
 - 6*a*b^5*c^5*d + 15*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 - 6*a^5*b*c*d^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*(256*b^6*d^6*x^5 + 15*b^6*c^5*d - 85*a*b^5*c^4*d^2 + 
198*a^2*b^4*c^3*d^3 + 198*a^3*b^3*c^2*d^4 - 85*a^4*b^2*c*d^5 + 15*a^5*b*d^ 
6 + 640*(b^6*c*d^5 + a*b^5*d^6)*x^4 + 16*(27*b^6*c^2*d^4 + 106*a*b^5*c*d^5 
 + 27*a^2*b^4*d^6)*x^3 + 8*(b^6*c^3*d^3 + 159*a*b^5*c^2*d^4 + 159*a^2*b^4* 
c*d^5 + a^3*b^3*d^6)*x^2 - 2*(5*b^6*c^4*d^2 - 28*a*b^5*c^3*d^3 - 594*a^2*b 
^4*c^2*d^4 - 28*a^3*b^3*c*d^5 + 5*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + 
 c))/(b^4*d^4)]
 

Sympy [F]

\[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\int \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \] Input:

integrate((b*x+a)**(5/2)*(d*x+c)**(5/2),x)
 

Output:

Integral((a + b*x)**(5/2)*(c + d*x)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3002 vs. \(2 (212) = 424\).

Time = 0.51 (sec) , antiderivative size = 3002, normalized size of antiderivative = 11.46 \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

-1/7680*(7680*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2 
*c + (b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b 
*d)*sqrt(b*x + a))*a^3*c^2*abs(b)/b^2 - 40*(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 - 9 
3*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(-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))*b*c^ 
2*abs(b) - 240*(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(-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*d*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*...
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx=\int {\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \] Input:

int((a + b*x)^(5/2)*(c + d*x)^(5/2),x)
 

Output:

int((a + b*x)^(5/2)*(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 860, normalized size of antiderivative = 3.28 \[ \int (a+b x)^{5/2} (c+d x)^{5/2} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)*(d*x+c)^(5/2),x)
 

Output:

(15*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b*d**6 - 85*sqrt(c + d*x)*sqrt(a + b* 
x)*a**4*b**2*c*d**5 - 10*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**2*d**6*x + 19 
8*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*c**2*d**4 + 56*sqrt(c + d*x)*sqrt( 
a + b*x)*a**3*b**3*c*d**5*x + 8*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**3*d**6 
*x**2 + 198*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*c**3*d**3 + 1188*sqrt(c 
+ d*x)*sqrt(a + b*x)*a**2*b**4*c**2*d**4*x + 1272*sqrt(c + d*x)*sqrt(a + b 
*x)*a**2*b**4*c*d**5*x**2 + 432*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**4*d**6 
*x**3 - 85*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*c**4*d**2 + 56*sqrt(c + d*x) 
*sqrt(a + b*x)*a*b**5*c**3*d**3*x + 1272*sqrt(c + d*x)*sqrt(a + b*x)*a*b** 
5*c**2*d**4*x**2 + 1696*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*c*d**5*x**3 + 6 
40*sqrt(c + d*x)*sqrt(a + b*x)*a*b**5*d**6*x**4 + 15*sqrt(c + d*x)*sqrt(a 
+ b*x)*b**6*c**5*d - 10*sqrt(c + d*x)*sqrt(a + b*x)*b**6*c**4*d**2*x + 8*s 
qrt(c + d*x)*sqrt(a + b*x)*b**6*c**3*d**3*x**2 + 432*sqrt(c + d*x)*sqrt(a 
+ b*x)*b**6*c**2*d**4*x**3 + 640*sqrt(c + d*x)*sqrt(a + b*x)*b**6*c*d**5*x 
**4 + 256*sqrt(c + d*x)*sqrt(a + b*x)*b**6*d**6*x**5 - 15*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 + 90*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d* 
x))/sqrt(a*d - b*c))*a**5*b*c*d**5 - 225*sqrt(d)*sqrt(b)*log((sqrt(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(c + d*x))...