3.14.50 \(\int (b d+2 c d x)^{3/2} (a+b x+c x^2)^{5/2} \, dx\) [1350]

Optimal. Leaf size=274 \[ -\frac {\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (b^2-4 a c\right )^{17/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{924 c^4 \sqrt {a+b x+c x^2}} \]

[Out]

-1/66*(-4*a*c+b^2)*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(3/2)/c^2/d+1/15*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(5/2)/
c/d+1/308*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-1/462*(-4*a*c+b^2)^3*d*(2*c*d*x+b*d)^(1
/2)*(c*x^2+b*x+a)^(1/2)/c^3-1/924*(-4*a*c+b^2)^(17/4)*d^(3/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)
/d^(1/2),I)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {699, 706, 705, 703, 227} \begin {gather*} -\frac {d^{3/2} \left (b^2-4 a c\right )^{17/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{924 c^4 \sqrt {a+b x+c x^2}}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{66 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{5/2}}{15 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

-1/462*((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/c^3 + ((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(5
/2)*Sqrt[a + b*x + c*x^2])/(308*c^3*d) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(66*c^2
*d) + ((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(5/2))/(15*c*d) - ((b^2 - 4*a*c)^(17/4)*d^(3/2)*Sqrt[-((c*(a +
b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(924*c
^4*Sqrt[a + b*x + c*x^2])

Rule 227

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

Rule 699

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((
a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x] - Dist[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] && NeQ[b^2 - 4*a*c, 0] && 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 703

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]

Rule 705

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && EqQ[m^2, 1/4]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && 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])

Rubi steps

\begin {align*} \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^2 \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx}{44 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (b^2-4 a c\right )^3 \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{616 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{1848 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{1848 c^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{924 c^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{462 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{66 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}}{15 c d}-\frac {\left (b^2-4 a c\right )^{17/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{924 c^4 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.09, size = 117, normalized size = 0.43 \begin {gather*} \frac {2}{15} d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^3+\frac {\left (b^2-4 a c\right )^3 \, _2F_1\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(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*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])))/15

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1054\) vs. \(2(234)=468\).
time = 0.87, size = 1055, normalized size = 3.85

method	result	size
risch	\(\frac {\left (1232 c^{6} x^{6}+3696 b \,c^{5} x^{5}+3584 a \,c^{5} x^{4}+3724 b^{2} c^{4} x^{4}+7168 a b \,c^{4} x^{3}+1288 b^{3} x^{3} c^{3}+3312 a^{2} c^{4} x^{2}+3720 a \,b^{2} c^{3} x^{2}+18 b^{4} c^{2} x^{2}+3312 a^{2} b \,c^{3} x +136 a \,b^{3} c^{2} x -10 b^{5} c x +640 a^{3} c^{3}+348 a^{2} b^{2} c^{2}-70 a \,b^{4} c +5 b^{6}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{2}}{4620 c^{3} \sqrt {d \left (2 c x +b \right )}}-\frac {\left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{924 c^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(642\)
default	\(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (-23712 a^{3} b \,c^{5} x^{2}-57104 a^{2} b^{2} c^{5} x^{3}-1280 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{3} b^{2} c^{3}+480 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{2} b^{4} c^{2}-80 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a \,b^{6} c -67424 a b \,c^{7} x^{6}-87344 a \,b^{2} c^{6} x^{5}-68960 a^{2} b \,c^{6} x^{4}-49800 a \,b^{3} c^{5} x^{4}-10624 a \,b^{4} c^{4} x^{3}-16696 a^{2} b^{3} c^{4} x^{2}+152 a \,b^{5} c^{3} x^{2}-9296 a^{3} b^{2} c^{4} x -688 a^{2} b^{4} c^{3} x +140 a \,b^{6} c^{2} x -4928 c^{9} x^{9}-2644 b^{5} c^{4} x^{4}-27584 a^{2} c^{7} x^{5}-34888 b^{3} c^{6} x^{6}-22176 b \,c^{8} x^{8}-19264 a \,c^{8} x^{7}-39536 b^{2} c^{7} x^{7}-15248 b^{4} c^{5} x^{5}-15808 a^{3} c^{6} x^{3}+4 b^{6} c^{3} x^{3}-10 b^{7} c^{2} x^{2}-2560 a^{4} c^{5} x -10 b^{8} c x -1280 a^{4} b \,c^{4}-696 a^{3} b^{3} c^{3}-10 a \,b^{7} c +5 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b^{8}+140 a^{2} b^{5} c^{2}+1280 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \EllipticF \left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, a^{4} c^{4}\right )}{9240 c^{4} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\)	\(1055\)
elliptic	\(\text {Expression too large to display}\)	\(5587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/9240*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d*(-23712*a^3*b*c^5*x^2-57104*a^2*b^2*c^5*x^3-67424*a*b*c^7*x^
6-87344*a*b^2*c^6*x^5-68960*a^2*b*c^6*x^4-49800*a*b^3*c^5*x^4-10624*a*b^4*c^4*x^3-16696*a^2*b^3*c^4*x^2+152*a*
b^5*c^3*x^2-9296*a^3*b^2*c^4*x-688*a^2*b^4*c^3*x+140*a*b^6*c^2*x-1280*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^
2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)
*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^3
*b^2*c^3+480*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-
b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)
^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a^2*b^4*c^2-80*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1
/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*Elli
pticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*a*b^6*c-
4928*c^9*x^9-2644*b^5*c^4*x^4-27584*a^2*c^7*x^5-34888*b^3*c^6*x^6+5*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)
^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*E
llipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*(-4*a*c+b^2)^(1/2)*b^8-2
2176*b*c^8*x^8-19264*a*c^8*x^7-39536*b^2*c^7*x^7-15248*b^4*c^5*x^5-15808*a^3*c^6*x^3+4*b^6*c^3*x^3-10*b^7*c^2*
x^2-2560*a^4*c^5*x-10*b^8*c*x-1280*a^4*b*c^4-696*a^3*b^3*c^3-10*a*b^7*c+140*a^2*b^5*c^2+1280*((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))
*(-4*a*c+b^2)^(1/2)*a^4*c^4)/c^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.88, size = 275, normalized size = 1.00 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (1232 \, c^{8} d x^{6} + 3696 \, b c^{7} d x^{5} + 28 \, {\left (133 \, b^{2} c^{6} + 128 \, a c^{7}\right )} d x^{4} + 56 \, {\left (23 \, b^{3} c^{5} + 128 \, a b c^{6}\right )} d x^{3} + 6 \, {\left (3 \, b^{4} c^{4} + 620 \, a b^{2} c^{5} + 552 \, a^{2} c^{6}\right )} d x^{2} - 2 \, {\left (5 \, b^{5} c^{3} - 68 \, a b^{3} c^{4} - 1656 \, a^{2} b c^{5}\right )} d x + {\left (5 \, b^{6} c^{2} - 70 \, a b^{4} c^{3} + 348 \, a^{2} b^{2} c^{4} + 640 \, a^{3} c^{5}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{9240 \, c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

-1/9240*(5*sqrt(2)*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(c^2*d)*d*weierstra
ssPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(1232*c^8*d*x^6 + 3696*b*c^7*d*x^5 + 28*(133*b^2*c^6 +
128*a*c^7)*d*x^4 + 56*(23*b^3*c^5 + 128*a*b*c^6)*d*x^3 + 6*(3*b^4*c^4 + 620*a*b^2*c^5 + 552*a^2*c^6)*d*x^2 -
2*(5*b^5*c^3 - 68*a*b^3*c^4 - 1656*a^2*b*c^5)*d*x + (5*b^6*c^2 - 70*a*b^4*c^3 + 348*a^2*b^2*c^4 + 640*a^3*c^5)
*d)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^5

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(3/2)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral((d*(b + 2*c*x))**(3/2)*(a + b*x + c*x**2)**(5/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(3/2)*(c*x^2 + b*x + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2), x)

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