\(\int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx\) [223]

Optimal result

Integrand size = 28, antiderivative size = 227 \[ \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{231 c^2 \sqrt {a+b x+c x^2}} \] Output:

-10/231*(-4*a*c+b^2)^2*d^3*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c-2/77* 
(-4*a*c+b^2)*d*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c+1/11*(2*c*d*x+b*d 
)^(9/2)*(c*x^2+b*x+a)^(1/2)/c/d-5/231*(-4*a*c+b^2)^(13/4)*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^2/(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 = 10.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.68 \[ \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx=\frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (7 (b+2 c x)^2 (a+x (b+c x))-10 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))+\frac {\left (b^2-4 a c\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{77 (b+2 c x)^3} \] Input:

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

Output:

(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(7*(b + 2*c*x)^2*(a + x*(b 
+ c*x)) - 10*(a - b^2/(4*c))*c*(2*(a + x*(b + c*x)) + ((b^2 - 4*a*c)*Hyper 
geometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c*Sqrt[(c*(a 
 + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(77*(b + 2*c*x)^3)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {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.

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

\(\Big \downarrow \) 1109

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1115

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

\(\Big \downarrow \) 1113

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

((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[ArcSin[Sq 
rt[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)
 

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. \(554\) vs. \(2(193)=386\).

Time = 2.64 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.44

method	result	size
risch	\(-\frac {\left (-336 c^{4} x^{4}-672 b \,c^{3} x^{3}-96 a \,c^{3} x^{2}-480 b^{2} c^{2} x^{2}-96 a b \,c^{2} x -144 b^{3} c x +160 a^{2} c^{2}-104 c a \,b^{2}-5 b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{4}}{231 c \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{231 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(555\)
default	\(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{3} \left (1344 c^{7} x^{7}+4704 b \,c^{6} x^{6}+1728 a \,c^{6} x^{5}+6624 b^{2} c^{5} x^{5}+320 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{3} c^{3}-240 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a^{2} b^{2} c^{2}+60 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a \,b^{4} c -5 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{6}+4320 a b \,c^{5} x^{4}+4800 b^{3} c^{4} x^{4}-256 a^{2} c^{5} x^{3}+4448 a \,b^{2} c^{4} x^{3}+1844 b^{4} c^{3} x^{3}-384 a^{2} b \,c^{4} x^{2}+2352 a \,b^{3} c^{3} x^{2}+318 b^{5} c^{2} x^{2}-640 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x +516 a \,b^{4} c^{2} x +10 b^{6} c x -320 a^{3} b \,c^{3}+208 a^{2} b^{3} c^{2}+10 a \,b^{5} c \right )}{462 c^{2} \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\)	\(798\)
elliptic	\(\text {Expression too large to display}\)	\(2307\)

Input:

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/231/c*(-336*c^4*x^4-672*b*c^3*x^3-96*a*c^3*x^2-480*b^2*c^2*x^2-96*a*b*c 
^2*x-144*b^3*c*x+160*a^2*c^2-104*a*b^2*c-5*b^4)*(c*x^2+b*x+a)^(1/2)*(2*c*x 
+b)*d^4/(d*(2*c*x+b))^(1/2)+5/231/c*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c- 
b^6)*(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)*EllipticF(((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 = 0.08 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.88 \[ \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx=-\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (336 \, c^{6} d^{3} x^{4} + 672 \, b c^{5} d^{3} x^{3} + 96 \, {\left (5 \, b^{2} c^{4} + a c^{5}\right )} d^{3} x^{2} + 48 \, {\left (3 \, b^{3} c^{3} + 2 \, a b c^{4}\right )} d^{3} x + {\left (5 \, b^{4} c^{2} + 104 \, a b^{2} c^{3} - 160 \, a^{2} c^{4}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{462 \, c^{3}} \] Input:

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

-1/462*(5*sqrt(2)*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c^ 
2*d)*d^3*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2* 
(336*c^6*d^3*x^4 + 672*b*c^5*d^3*x^3 + 96*(5*b^2*c^4 + a*c^5)*d^3*x^2 + 48 
*(3*b^3*c^3 + 2*a*b*c^4)*d^3*x + (5*b^4*c^2 + 104*a*b^2*c^3 - 160*a^2*c^4) 
*d^3)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^3
 

Sympy [F]

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

integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2),x)
 

Output:

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2), x)
 

Reduce [F]

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

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x)
 

Output:

(sqrt(d)*d**3*( - 64*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b**2*c + 
64*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a**2*b*c**2*x + 64*sqrt(b + 2*c* 
x)*sqrt(a + b*x + c*x**2)*a**2*c**3*x**2 + 58*sqrt(b + 2*c*x)*sqrt(a + b*x 
 + c*x**2)*a*b**4 + 128*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**3*c*x 
+ 352*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b**2*c**2*x**2 + 448*sqrt(b 
 + 2*c*x)*sqrt(a + b*x + c*x**2)*a*b*c**3*x**3 + 224*sqrt(b + 2*c*x)*sqrt( 
a + b*x + c*x**2)*a*c**4*x**4 + 48*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)* 
b**5*x + 160*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**4*c*x**2 + 224*sqrt 
(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**3*c**2*x**3 + 112*sqrt(b + 2*c*x)*sq 
rt(a + b*x + c*x**2)*b**2*c**3*x**4 - 640*int((sqrt(b + 2*c*x)*sqrt(a + b* 
x + c*x**2)*x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6* 
a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3) 
,x)*a**4*c**5 + 160*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a 
**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c* 
*3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a**3*b**2*c**4 + 1 
20*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4*a**2* 
c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x 
+ 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a**2*b**4*c**3 - 50*int((sqrt(b + 2 
*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 
4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2...