\(\int (b d+2 c d x)^{7/2} (a+b x+c x^2)^{5/2} \, dx\) [245]

Optimal result

Integrand size = 28, antiderivative size = 321 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right )^{21/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{17556 c^4 \sqrt {a+b x+c x^2}} \] Output:

-5/8778*(-4*a*c+b^2)^4*d^3*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3-1/2 
926*(-4*a*c+b^2)^3*d*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c^3+1/836*(-4 
*a*c+b^2)^2*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-1/114*(-4*a*c+b^ 
2)*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(3/2)/c^2/d+1/19*(2*c*d*x+b*d)^(9/2)* 
(c*x^2+b*x+a)^(5/2)/c/d-5/17556*(-4*a*c+b^2)^(21/4)*d^(7/2)*(-c*(c*x^2+b*x 
+a)/(-4*a*c+b^2))^(1/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d 
^(1/2),I)/c^4/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.50 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (3 (b+2 c x)^2 (a+x (b+c x))^3-2 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))^3+\frac {\left (b^2-4 a c\right )^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{57 (b+2 c x)^3} \] Input:

Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(3*(b + 2*c*x)^2*(a + x*(b 
+ c*x))^3 - 2*(a - b^2/(4*c))*c*(2*(a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3* 
Hypergeometric2F1[-5/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(64*c^3*Sq 
rt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(57*(b + 2*c*x)^3)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1109, 1109, 1109, 1116, 1116, 1115, 1113, 762}

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.

\(\displaystyle \int \left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2} \, dx\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \int (b d+2 c x d)^{7/2} \left (c x^2+b x+a\right )^{3/2}dx}{38 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c x d)^{7/2} \sqrt {c x^2+b x+a}dx}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{7/2}}{\sqrt {c x^2+b x+a}}dx}{22 c}\right )}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \int \frac {(b d+2 c x d)^{3/2}}{\sqrt {c x^2+b x+a}}dx+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )}{22 c}\right )}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {1}{3} d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )}{22 c}\right )}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 1115

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {d^2 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )}{22 c}\right )}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 1113

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {2 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )}{22 c}\right )}{10 c}\right )}{38 c}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \left (\frac {5}{7} d^2 \left (b^2-4 a c\right ) \left (\frac {2 d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\right )+\frac {4}{7} d \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}\right )}{22 c}\right )}{10 c}\right )}{38 c}\)

Input:

Int[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

((b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(5/2))/(19*c*d) - (5*(b^2 - 4*a*c 
)*(((b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(15*c*d) - ((b^2 - 4*a* 
c)*(((b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(11*c*d) - ((b^2 - 4*a*c 
)*((4*d*(b*d + 2*c*d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/7 + (5*(b^2 - 4*a*c)* 
d^2*((4*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/3 + (2*(b^2 - 4*a*c)^ 
(5/4)*d^(3/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSi 
n[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c*Sqrt[a + b 
*x + c*x^2])))/7))/(22*c)))/(10*c)))/(38*c)
 

Defintions of rubi rules used

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 1109

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

rule 1113

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ 
Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)]   Subst[Int[1/Sqrt[Simp[1 - b^ 
2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, 
 c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
 

rule 1115

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym 
bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* 
x^2]   Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) 
- c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* 
d - b*e, 0] && EqQ[m^2, 1/4]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(760\) vs. \(2(275)=550\).

Time = 9.92 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.37

method	result	size
risch	\(-\frac {\left (-14784 x^{8} c^{8}-59136 b \,c^{7} x^{7}-39424 a \,c^{7} x^{6}-93632 b^{2} c^{6} x^{6}-118272 a b \,c^{6} x^{5}-73920 b^{3} c^{5} x^{5}-30016 a^{2} c^{6} x^{4}-132832 a \,b^{2} c^{5} x^{4}-29596 b^{4} c^{4} x^{4}-60032 a^{2} b \,c^{5} x^{3}-68544 a \,b^{3} c^{4} x^{3}-4984 b^{5} c^{3} x^{3}-1536 a^{3} c^{5} x^{2}-43872 a^{2} b^{2} c^{4} x^{2}-14736 a \,b^{4} c^{3} x^{2}-18 b^{6} c^{2} x^{2}-1536 a^{3} b \,c^{4} x -13856 a^{2} b^{3} c^{3} x -176 a \,b^{5} c^{2} x +10 b^{7} c x +2560 a^{4} c^{4}-2944 a^{3} b^{2} c^{3}-628 a^{2} b^{4} c^{2}+90 a \,b^{6} c -5 b^{8}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{4}}{17556 c^{3} \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (1024 a^{5} c^{5}-1280 a^{4} b^{2} c^{4}+640 a^{3} b^{4} c^{3}-160 a^{2} b^{6} c^{2}+20 a \,b^{8} c -b^{10}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{17556 c^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(761\)
default	\(\text {Expression too large to display}\)	\(1344\)
elliptic	\(\text {Expression too large to display}\)	\(17392\)

Input:

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/17556/c^3*(-14784*c^8*x^8-59136*b*c^7*x^7-39424*a*c^7*x^6-93632*b^2*c^6 
*x^6-118272*a*b*c^6*x^5-73920*b^3*c^5*x^5-30016*a^2*c^6*x^4-132832*a*b^2*c 
^5*x^4-29596*b^4*c^4*x^4-60032*a^2*b*c^5*x^3-68544*a*b^3*c^4*x^3-4984*b^5* 
c^3*x^3-1536*a^3*c^5*x^2-43872*a^2*b^2*c^4*x^2-14736*a*b^4*c^3*x^2-18*b^6* 
c^2*x^2-1536*a^3*b*c^4*x-13856*a^2*b^3*c^3*x-176*a*b^5*c^2*x+10*b^7*c*x+25 
60*a^4*c^4-2944*a^3*b^2*c^3-628*a^2*b^4*c^2+90*a*b^6*c-5*b^8)*(c*x^2+b*x+a 
)^(1/2)*(2*c*x+b)*d^4/(d*(2*c*x+b))^(1/2)+5/17556/c^3*(1024*a^5*c^5-1280*a 
^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)*(1/2/c*(-b+(-4 
*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+1/2*(b+(-4*a*c+b^2)^(1/ 
2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2) 
*((x+1/2*b/c)/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c))^(1/2)*((x-1/2/c*(-b 
+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2 
)^(1/2))))^(1/2)/(2*c^2*d*x^3+3*b*c*d*x^2+2*a*c*d*x+b^2*d*x+a*b*d)^(1/2)*E 
llipticF(((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(1/2/c*(-b+(-4*a*c+b^2)^(1/2))+ 
1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c 
*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+1/2*b/c))^(1/2))* 
d^4*(d*(2*c*x+b)*(c*x^2+b*x+a))^(1/2)/(d*(2*c*x+b))^(1/2)/(c*x^2+b*x+a)^(1 
/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.26 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \, \sqrt {2} {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (14784 \, c^{10} d^{3} x^{8} + 59136 \, b c^{9} d^{3} x^{7} + 4928 \, {\left (19 \, b^{2} c^{8} + 8 \, a c^{9}\right )} d^{3} x^{6} + 14784 \, {\left (5 \, b^{3} c^{7} + 8 \, a b c^{8}\right )} d^{3} x^{5} + 28 \, {\left (1057 \, b^{4} c^{6} + 4744 \, a b^{2} c^{7} + 1072 \, a^{2} c^{8}\right )} d^{3} x^{4} + 56 \, {\left (89 \, b^{5} c^{5} + 1224 \, a b^{3} c^{6} + 1072 \, a^{2} b c^{7}\right )} d^{3} x^{3} + 6 \, {\left (3 \, b^{6} c^{4} + 2456 \, a b^{4} c^{5} + 7312 \, a^{2} b^{2} c^{6} + 256 \, a^{3} c^{7}\right )} d^{3} x^{2} - 2 \, {\left (5 \, b^{7} c^{3} - 88 \, a b^{5} c^{4} - 6928 \, a^{2} b^{3} c^{5} - 768 \, a^{3} b c^{6}\right )} d^{3} x + {\left (5 \, b^{8} c^{2} - 90 \, a b^{6} c^{3} + 628 \, a^{2} b^{4} c^{4} + 2944 \, a^{3} b^{2} c^{5} - 2560 \, a^{4} c^{6}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{35112 \, c^{5}} \] Input:

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

-1/35112*(5*sqrt(2)*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 
 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(c^2*d)*d^3*weierstrassPInverse((b 
^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(14784*c^10*d^3*x^8 + 59136*b*c 
^9*d^3*x^7 + 4928*(19*b^2*c^8 + 8*a*c^9)*d^3*x^6 + 14784*(5*b^3*c^7 + 8*a* 
b*c^8)*d^3*x^5 + 28*(1057*b^4*c^6 + 4744*a*b^2*c^7 + 1072*a^2*c^8)*d^3*x^4 
 + 56*(89*b^5*c^5 + 1224*a*b^3*c^6 + 1072*a^2*b*c^7)*d^3*x^3 + 6*(3*b^6*c^ 
4 + 2456*a*b^4*c^5 + 7312*a^2*b^2*c^6 + 256*a^3*c^7)*d^3*x^2 - 2*(5*b^7*c^ 
3 - 88*a*b^5*c^4 - 6928*a^2*b^3*c^5 - 768*a^3*b*c^6)*d^3*x + (5*b^8*c^2 - 
90*a*b^6*c^3 + 628*a^2*b^4*c^4 + 2944*a^3*b^2*c^5 - 2560*a^4*c^6)*d^3)*sqr 
t(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^5
 

Sympy [F]

\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral((d*(b + 2*c*x))**(7/2)*(a + b*x + c*x**2)**(5/2), x)
 

Maxima [F]

\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(5/2), x)
 

Giac [F]

\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\text {too large to display} \] Input:

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

(sqrt(d)*d**3*( - 3072*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**4*b**2*c* 
*3 + 3072*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**4*b*c**4*x + 3072*sqrt 
(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**4*c**5*x**2 + 7400*sqrt(b + 2*c*x)*s 
qrt(a + b*x + c*x**2)*a**3*b**4*c**2 + 29248*sqrt(b + 2*c*x)*sqrt(a + b*x 
+ c*x**2)*a**3*b**3*c**3*x + 89280*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)* 
a**3*b**2*c**4*x**2 + 120064*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**3*b 
*c**5*x**3 + 60032*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**3*c**6*x**4 - 
 352*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**6*c + 14208*sqrt(b + 2 
*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**5*c**2*x + 73344*sqrt(b + 2*c*x)*sqrt 
(a + b*x + c*x**2)*a**2*b**4*c**3*x**2 + 197120*sqrt(b + 2*c*x)*sqrt(a + b 
*x + c*x**2)*a**2*b**3*c**4*x**3 + 295680*sqrt(b + 2*c*x)*sqrt(a + b*x + c 
*x**2)*a**2*b**2*c**5*x**4 + 236544*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2) 
*a**2*b*c**6*x**5 + 78848*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*c**7 
*x**6 + 20*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**8 + 156*sqrt(b + 2* 
c*x)*sqrt(a + b*x + c*x**2)*a*b**7*c*x + 14772*sqrt(b + 2*c*x)*sqrt(a + b* 
x + c*x**2)*a*b**6*c**2*x**2 + 78512*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2 
)*a*b**5*c**3*x**3 + 192024*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**4* 
c**4*x**4 + 266112*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**3*c**5*x**5 
 + 226688*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2*c**6*x**6 + 118272 
*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b*c**7*x**7 + 29568*sqrt(b + ...