\(\int (b d+2 c d x)^5 (a+b x+c x^2)^{5/2} \, dx\) [177]

Optimal result

Integrand size = 26, antiderivative size = 98 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {16}{693} \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{7/2}+\frac {8}{99} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2} \] Output:

16/693*(-4*a*c+b^2)^2*d^5*(c*x^2+b*x+a)^(7/2)+8/99*(-4*a*c+b^2)*d^5*(2*c*x 
+b)^2*(c*x^2+b*x+a)^(7/2)+2/11*d^5*(2*c*x+b)^4*(c*x^2+b*x+a)^(7/2)
 

Mathematica [A] (verified)

Time = 3.72 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{693} d^5 (a+x (b+c x))^{7/2} \left (99 b^4+616 b^3 c x+224 b c^2 x \left (-2 a+9 c x^2\right )+8 b^2 c \left (-22 a+203 c x^2\right )+16 c^2 \left (8 a^2-28 a c x^2+63 c^2 x^4\right )\right ) \] Input:

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

Output:

(2*d^5*(a + x*(b + c*x))^(7/2)*(99*b^4 + 616*b^3*c*x + 224*b*c^2*x*(-2*a + 
 9*c*x^2) + 8*b^2*c*(-22*a + 203*c*x^2) + 16*c^2*(8*a^2 - 28*a*c*x^2 + 63* 
c^2*x^4)))/693
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1116, 27, 1116, 1104}

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.

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

\(\Big \downarrow \) 1116

\(\displaystyle \frac {4}{11} d^2 \left (b^2-4 a c\right ) \int d^3 (b+2 c x)^3 \left (c x^2+b x+a\right )^{5/2}dx+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{11} d^5 \left (b^2-4 a c\right ) \int (b+2 c x)^3 \left (c x^2+b x+a\right )^{5/2}dx+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {4}{11} d^5 \left (b^2-4 a c\right ) \left (\frac {2}{9} \left (b^2-4 a c\right ) \int (b+2 c x) \left (c x^2+b x+a\right )^{5/2}dx+\frac {2}{9} (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}\right )+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}\)

\(\Big \downarrow \) 1104

\(\displaystyle \frac {4}{11} d^5 \left (b^2-4 a c\right ) \left (\frac {4}{63} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac {2}{9} (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}\right )+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}\)

Input:

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

Output:

(2*d^5*(b + 2*c*x)^4*(a + b*x + c*x^2)^(7/2))/11 + (4*(b^2 - 4*a*c)*d^5*(( 
4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(7/2))/63 + (2*(b + 2*c*x)^2*(a + b*x + 
c*x^2)^(7/2))/9))/11
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1104

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol 
] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, 
d, e, p}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1116

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p 
 + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p 
+ 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
 

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81

method	result	size
pseudoelliptic	\(\frac {256 \left (\frac {63 c^{4} x^{4}}{8}-\frac {7 x^{2} \left (-\frac {9 b x}{2}+a \right ) c^{3}}{2}+\left (\frac {203}{16} b^{2} x^{2}-\frac {7}{2} a b x +a^{2}\right ) c^{2}-\frac {11 b^{2} \left (-\frac {7 b x}{2}+a \right ) c}{8}+\frac {99 b^{4}}{128}\right ) d^{5} \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{693}\)	\(79\)
gosper	\(\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (1008 c^{4} x^{4}+2016 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+1624 b^{2} c^{2} x^{2}-448 a b \,c^{2} x +616 b^{3} c x +128 a^{2} c^{2}-176 c a \,b^{2}+99 b^{4}\right ) d^{5}}{693}\)	\(91\)
orering	\(\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (1008 c^{4} x^{4}+2016 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+1624 b^{2} c^{2} x^{2}-448 a b \,c^{2} x +616 b^{3} c x +128 a^{2} c^{2}-176 c a \,b^{2}+99 b^{4}\right ) \left (2 c d x +b d \right )^{5}}{693 \left (2 c x +b \right )^{5}}\)	\(107\)
trager	\(d^{5} \left (\frac {2254}{99} a \,b^{4} c^{2} x^{4}+\frac {412}{63} a \,b^{5} c \,x^{3}+\frac {14}{3} a^{2} b^{4} c \,x^{2}+\frac {16}{63} a^{3} b^{3} c x +\frac {2}{7} a^{3} b^{4}+\frac {2944}{99} c^{5} b a \,x^{7}+\frac {5890}{231} b^{4} c^{3} x^{6}-\frac {128}{693} c^{2} a^{4} b x +\frac {272}{693} a^{3} b^{2} c^{2} x^{2}+\frac {64}{231} a^{3} b \,c^{3} x^{3}+\frac {8896}{693} a^{2} b^{3} c^{2} x^{3}+\frac {34256}{693} a \,b^{2} c^{4} x^{6}+\frac {4496}{231} a^{2} b^{2} c^{3} x^{4}+\frac {30640}{693} a \,b^{3} c^{3} x^{5}+\frac {3616}{231} a^{2} b \,c^{4} x^{5}+\frac {736}{99} a \,c^{6} x^{8}+\frac {3056}{99} c^{5} b^{2} x^{8}+\frac {256}{693} a^{5} c^{2}+\frac {3584}{99} b^{3} c^{4} x^{7}-\frac {128}{693} a^{4} c^{3} x^{2}+\frac {32}{231} a^{3} c^{4} x^{4}+\frac {3616}{693} a^{2} c^{5} x^{6}+\frac {6}{7} a \,b^{6} x^{2}+\frac {7538}{693} b^{5} c^{2} x^{5}+\frac {2}{7} b^{7} x^{3}+\frac {32}{11} c^{7} x^{10}+\frac {166}{63} b^{6} c \,x^{4}+\frac {6}{7} a^{2} b^{5} x -\frac {32}{63} a^{4} b^{2} c +\frac {160}{11} b \,c^{6} x^{9}\right ) \sqrt {c \,x^{2}+b x +a}\)	\(335\)
risch	\(\frac {2 d^{5} \left (1008 c^{7} x^{10}+5040 b \,c^{6} x^{9}+2576 a \,c^{6} x^{8}+10696 c^{5} b^{2} x^{8}+10304 c^{5} b a \,x^{7}+12544 b^{3} c^{4} x^{7}+1808 a^{2} c^{5} x^{6}+17128 a \,b^{2} c^{4} x^{6}+8835 b^{4} c^{3} x^{6}+5424 a^{2} b \,c^{4} x^{5}+15320 a \,b^{3} c^{3} x^{5}+3769 b^{5} c^{2} x^{5}+48 a^{3} c^{4} x^{4}+6744 a^{2} b^{2} c^{3} x^{4}+7889 a \,b^{4} c^{2} x^{4}+913 b^{6} c \,x^{4}+96 a^{3} b \,c^{3} x^{3}+4448 a^{2} b^{3} c^{2} x^{3}+2266 a \,b^{5} c \,x^{3}+99 b^{7} x^{3}-64 a^{4} c^{3} x^{2}+136 a^{3} b^{2} c^{2} x^{2}+1617 a^{2} b^{4} c \,x^{2}+297 a \,b^{6} x^{2}-64 c^{2} a^{4} b x +88 a^{3} b^{3} c x +297 a^{2} b^{5} x +128 a^{5} c^{2}-176 a^{4} b^{2} c +99 a^{3} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{693}\)	\(336\)
default	\(\text {Expression too large to display}\)	\(3619\)

Input:

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

Output:

256/693*(63/8*c^4*x^4-7/2*x^2*(-9/2*b*x+a)*c^3+(203/16*b^2*x^2-7/2*a*b*x+a 
^2)*c^2-11/8*b^2*(-7/2*b*x+a)*c+99/128*b^4)*d^5*(c*x^2+b*x+a)^(7/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (86) = 172\).

Time = 0.14 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.45 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{693} \, {\left (1008 \, c^{7} d^{5} x^{10} + 5040 \, b c^{6} d^{5} x^{9} + 56 \, {\left (191 \, b^{2} c^{5} + 46 \, a c^{6}\right )} d^{5} x^{8} + 448 \, {\left (28 \, b^{3} c^{4} + 23 \, a b c^{5}\right )} d^{5} x^{7} + {\left (8835 \, b^{4} c^{3} + 17128 \, a b^{2} c^{4} + 1808 \, a^{2} c^{5}\right )} d^{5} x^{6} + {\left (3769 \, b^{5} c^{2} + 15320 \, a b^{3} c^{3} + 5424 \, a^{2} b c^{4}\right )} d^{5} x^{5} + {\left (913 \, b^{6} c + 7889 \, a b^{4} c^{2} + 6744 \, a^{2} b^{2} c^{3} + 48 \, a^{3} c^{4}\right )} d^{5} x^{4} + {\left (99 \, b^{7} + 2266 \, a b^{5} c + 4448 \, a^{2} b^{3} c^{2} + 96 \, a^{3} b c^{3}\right )} d^{5} x^{3} + {\left (297 \, a b^{6} + 1617 \, a^{2} b^{4} c + 136 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} d^{5} x^{2} + {\left (297 \, a^{2} b^{5} + 88 \, a^{3} b^{3} c - 64 \, a^{4} b c^{2}\right )} d^{5} x + {\left (99 \, a^{3} b^{4} - 176 \, a^{4} b^{2} c + 128 \, a^{5} c^{2}\right )} d^{5}\right )} \sqrt {c x^{2} + b x + a} \] Input:

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

Output:

2/693*(1008*c^7*d^5*x^10 + 5040*b*c^6*d^5*x^9 + 56*(191*b^2*c^5 + 46*a*c^6 
)*d^5*x^8 + 448*(28*b^3*c^4 + 23*a*b*c^5)*d^5*x^7 + (8835*b^4*c^3 + 17128* 
a*b^2*c^4 + 1808*a^2*c^5)*d^5*x^6 + (3769*b^5*c^2 + 15320*a*b^3*c^3 + 5424 
*a^2*b*c^4)*d^5*x^5 + (913*b^6*c + 7889*a*b^4*c^2 + 6744*a^2*b^2*c^3 + 48* 
a^3*c^4)*d^5*x^4 + (99*b^7 + 2266*a*b^5*c + 4448*a^2*b^3*c^2 + 96*a^3*b*c^ 
3)*d^5*x^3 + (297*a*b^6 + 1617*a^2*b^4*c + 136*a^3*b^2*c^2 - 64*a^4*c^3)*d 
^5*x^2 + (297*a^2*b^5 + 88*a^3*b^3*c - 64*a^4*b*c^2)*d^5*x + (99*a^3*b^4 - 
 176*a^4*b^2*c + 128*a^5*c^2)*d^5)*sqrt(c*x^2 + b*x + a)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 913 vs. \(2 (97) = 194\).

Time = 0.40 (sec) , antiderivative size = 913, normalized size of antiderivative = 9.32 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {256 a^{5} c^{2} d^{5} \sqrt {a + b x + c x^{2}}}{693} - \frac {32 a^{4} b^{2} c d^{5} \sqrt {a + b x + c x^{2}}}{63} - \frac {128 a^{4} b c^{2} d^{5} x \sqrt {a + b x + c x^{2}}}{693} - \frac {128 a^{4} c^{3} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{693} + \frac {2 a^{3} b^{4} d^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {16 a^{3} b^{3} c d^{5} x \sqrt {a + b x + c x^{2}}}{63} + \frac {272 a^{3} b^{2} c^{2} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{693} + \frac {64 a^{3} b c^{3} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{231} + \frac {32 a^{3} c^{4} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{231} + \frac {6 a^{2} b^{5} d^{5} x \sqrt {a + b x + c x^{2}}}{7} + \frac {14 a^{2} b^{4} c d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{3} + \frac {8896 a^{2} b^{3} c^{2} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{693} + \frac {4496 a^{2} b^{2} c^{3} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{231} + \frac {3616 a^{2} b c^{4} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{231} + \frac {3616 a^{2} c^{5} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{693} + \frac {6 a b^{6} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {412 a b^{5} c d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{63} + \frac {2254 a b^{4} c^{2} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{99} + \frac {30640 a b^{3} c^{3} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{693} + \frac {34256 a b^{2} c^{4} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{693} + \frac {2944 a b c^{5} d^{5} x^{7} \sqrt {a + b x + c x^{2}}}{99} + \frac {736 a c^{6} d^{5} x^{8} \sqrt {a + b x + c x^{2}}}{99} + \frac {2 b^{7} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {166 b^{6} c d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{63} + \frac {7538 b^{5} c^{2} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{693} + \frac {5890 b^{4} c^{3} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{231} + \frac {3584 b^{3} c^{4} d^{5} x^{7} \sqrt {a + b x + c x^{2}}}{99} + \frac {3056 b^{2} c^{5} d^{5} x^{8} \sqrt {a + b x + c x^{2}}}{99} + \frac {160 b c^{6} d^{5} x^{9} \sqrt {a + b x + c x^{2}}}{11} + \frac {32 c^{7} d^{5} x^{10} \sqrt {a + b x + c x^{2}}}{11} \] Input:

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

Output:

256*a**5*c**2*d**5*sqrt(a + b*x + c*x**2)/693 - 32*a**4*b**2*c*d**5*sqrt(a 
 + b*x + c*x**2)/63 - 128*a**4*b*c**2*d**5*x*sqrt(a + b*x + c*x**2)/693 - 
128*a**4*c**3*d**5*x**2*sqrt(a + b*x + c*x**2)/693 + 2*a**3*b**4*d**5*sqrt 
(a + b*x + c*x**2)/7 + 16*a**3*b**3*c*d**5*x*sqrt(a + b*x + c*x**2)/63 + 2 
72*a**3*b**2*c**2*d**5*x**2*sqrt(a + b*x + c*x**2)/693 + 64*a**3*b*c**3*d* 
*5*x**3*sqrt(a + b*x + c*x**2)/231 + 32*a**3*c**4*d**5*x**4*sqrt(a + b*x + 
 c*x**2)/231 + 6*a**2*b**5*d**5*x*sqrt(a + b*x + c*x**2)/7 + 14*a**2*b**4* 
c*d**5*x**2*sqrt(a + b*x + c*x**2)/3 + 8896*a**2*b**3*c**2*d**5*x**3*sqrt( 
a + b*x + c*x**2)/693 + 4496*a**2*b**2*c**3*d**5*x**4*sqrt(a + b*x + c*x** 
2)/231 + 3616*a**2*b*c**4*d**5*x**5*sqrt(a + b*x + c*x**2)/231 + 3616*a**2 
*c**5*d**5*x**6*sqrt(a + b*x + c*x**2)/693 + 6*a*b**6*d**5*x**2*sqrt(a + b 
*x + c*x**2)/7 + 412*a*b**5*c*d**5*x**3*sqrt(a + b*x + c*x**2)/63 + 2254*a 
*b**4*c**2*d**5*x**4*sqrt(a + b*x + c*x**2)/99 + 30640*a*b**3*c**3*d**5*x* 
*5*sqrt(a + b*x + c*x**2)/693 + 34256*a*b**2*c**4*d**5*x**6*sqrt(a + b*x + 
 c*x**2)/693 + 2944*a*b*c**5*d**5*x**7*sqrt(a + b*x + c*x**2)/99 + 736*a*c 
**6*d**5*x**8*sqrt(a + b*x + c*x**2)/99 + 2*b**7*d**5*x**3*sqrt(a + b*x + 
c*x**2)/7 + 166*b**6*c*d**5*x**4*sqrt(a + b*x + c*x**2)/63 + 7538*b**5*c** 
2*d**5*x**5*sqrt(a + b*x + c*x**2)/693 + 5890*b**4*c**3*d**5*x**6*sqrt(a + 
 b*x + c*x**2)/231 + 3584*b**3*c**4*d**5*x**7*sqrt(a + b*x + c*x**2)/99 + 
3056*b**2*c**5*d**5*x**8*sqrt(a + b*x + c*x**2)/99 + 160*b*c**6*d**5*x*...
 

Maxima [F(-2)]

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

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} b^{4} d^{5} + \frac {16}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} b^{2} c d^{5} - \frac {16}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a b^{2} c d^{5} + \frac {32}{11} \, {\left (c x^{2} + b x + a\right )}^{\frac {11}{2}} c^{2} d^{5} - \frac {64}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} a c^{2} d^{5} + \frac {32}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a^{2} c^{2} d^{5} \] Input:

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

Output:

2/7*(c*x^2 + b*x + a)^(7/2)*b^4*d^5 + 16/9*(c*x^2 + b*x + a)^(9/2)*b^2*c*d 
^5 - 16/7*(c*x^2 + b*x + a)^(7/2)*a*b^2*c*d^5 + 32/11*(c*x^2 + b*x + a)^(1 
1/2)*c^2*d^5 - 64/9*(c*x^2 + b*x + a)^(9/2)*a*c^2*d^5 + 32/7*(c*x^2 + b*x 
+ a)^(7/2)*a^2*c^2*d^5
 

Mupad [B] (verification not implemented)

Time = 6.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2\,b^4\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}+\frac {32\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{11/2}}{11}-\frac {64\,a\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{9/2}}{9}+\frac {16\,b^2\,c\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{9/2}}{9}+\frac {32\,a^2\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}-\frac {16\,a\,b^2\,c\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \] Input:

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

Output:

(2*b^4*d^5*(a + b*x + c*x^2)^(7/2))/7 + (32*c^2*d^5*(a + b*x + c*x^2)^(11/ 
2))/11 - (64*a*c^2*d^5*(a + b*x + c*x^2)^(9/2))/9 + (16*b^2*c*d^5*(a + b*x 
 + c*x^2)^(9/2))/9 + (32*a^2*c^2*d^5*(a + b*x + c*x^2)^(7/2))/7 - (16*a*b^ 
2*c*d^5*(a + b*x + c*x^2)^(7/2))/7
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.41 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {c \,x^{2}+b x +a}\, d^{5} \left (1008 c^{7} x^{10}+5040 b \,c^{6} x^{9}+2576 a \,c^{6} x^{8}+10696 b^{2} c^{5} x^{8}+10304 a b \,c^{5} x^{7}+12544 b^{3} c^{4} x^{7}+1808 a^{2} c^{5} x^{6}+17128 a \,b^{2} c^{4} x^{6}+8835 b^{4} c^{3} x^{6}+5424 a^{2} b \,c^{4} x^{5}+15320 a \,b^{3} c^{3} x^{5}+3769 b^{5} c^{2} x^{5}+48 a^{3} c^{4} x^{4}+6744 a^{2} b^{2} c^{3} x^{4}+7889 a \,b^{4} c^{2} x^{4}+913 b^{6} c \,x^{4}+96 a^{3} b \,c^{3} x^{3}+4448 a^{2} b^{3} c^{2} x^{3}+2266 a \,b^{5} c \,x^{3}+99 b^{7} x^{3}-64 a^{4} c^{3} x^{2}+136 a^{3} b^{2} c^{2} x^{2}+1617 a^{2} b^{4} c \,x^{2}+297 a \,b^{6} x^{2}-64 a^{4} b \,c^{2} x +88 a^{3} b^{3} c x +297 a^{2} b^{5} x +128 a^{5} c^{2}-176 a^{4} b^{2} c +99 a^{3} b^{4}\right )}{693} \] Input:

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

Output:

(2*sqrt(a + b*x + c*x**2)*d**5*(128*a**5*c**2 - 176*a**4*b**2*c - 64*a**4* 
b*c**2*x - 64*a**4*c**3*x**2 + 99*a**3*b**4 + 88*a**3*b**3*c*x + 136*a**3* 
b**2*c**2*x**2 + 96*a**3*b*c**3*x**3 + 48*a**3*c**4*x**4 + 297*a**2*b**5*x 
 + 1617*a**2*b**4*c*x**2 + 4448*a**2*b**3*c**2*x**3 + 6744*a**2*b**2*c**3* 
x**4 + 5424*a**2*b*c**4*x**5 + 1808*a**2*c**5*x**6 + 297*a*b**6*x**2 + 226 
6*a*b**5*c*x**3 + 7889*a*b**4*c**2*x**4 + 15320*a*b**3*c**3*x**5 + 17128*a 
*b**2*c**4*x**6 + 10304*a*b*c**5*x**7 + 2576*a*c**6*x**8 + 99*b**7*x**3 + 
913*b**6*c*x**4 + 3769*b**5*c**2*x**5 + 8835*b**4*c**3*x**6 + 12544*b**3*c 
**4*x**7 + 10696*b**2*c**5*x**8 + 5040*b*c**6*x**9 + 1008*c**7*x**10))/693