3.14.37 \(\int (b d+2 c d x)^{7/2} (a+b x+c x^2)^{3/2} \, dx\) [1337]

Optimal. Leaf size=274 \[ \frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^{17/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{462 c^3 \sqrt {a+b x+c x^2}} \]

[Out]

1/15*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(3/2)/c/d+1/385*(-4*a*c+b^2)^2*d*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2
)/c^2-1/110*(-4*a*c+b^2)*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(1/2)/c^2/d+1/231*(-4*a*c+b^2)^3*d^3*(2*c*d*x+b*d)^
(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+1/462*(-4*a*c+b^2)^(17/4)*d^(7/2)*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^3/(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^{7/2} \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 )}{462 c^3 \sqrt {a+b x+c x^2}}+\frac {d^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c^2}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac {d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

((b^2 - 4*a*c)^3*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c^2) + ((b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)
^(5/2)*Sqrt[a + b*x + c*x^2])/(385*c^2) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(110*c^2
*d) + ((b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(15*c*d) + ((b^2 - 4*a*c)^(17/4)*d^(7/2)*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])/(462*c
^3*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)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx}{10 c}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{220 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{308 c^2}\\ &=\frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{924 c^2}\\ &=\frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^4 \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}{924 c^2 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^3 \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 )}{462 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^{17/4} d^{7/2} \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 )}{462 c^3 \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 = 10.30, size = 161, normalized size = 0.59 \begin {gather*} \frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (11 (b+2 c x)^2 (a+x (b+c x))^2-10 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))^2-\frac {\left (b^2-4 a c\right )^2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{165 (b+2 c x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x]

[Out]

(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(11*(b + 2*c*x)^2*(a + x*(b + c*x))^2 - 10*(a - b^2/(4*c))*c*(2
*(a + x*(b + c*x))^2 - ((b^2 - 4*a*c)^2*Hypergeometric2F1[-3/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(16*c^
2*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(165*(b + 2*c*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1056\) vs. \(2(234)=468\).
time = 0.79, size = 1057, normalized size = 3.86

method	result	size
risch	\(-\frac {\left (-2464 c^{6} x^{6}-7392 b \,c^{5} x^{5}-3808 a \,c^{5} x^{4}-8288 b^{2} c^{4} x^{4}-7616 a b \,c^{4} x^{3}-4256 b^{3} x^{3} c^{3}-384 a^{2} c^{4} x^{2}-5520 a \,b^{2} c^{3} x^{2}-906 b^{4} c^{2} x^{2}-384 a^{2} b \,c^{3} x -1712 a \,b^{3} c^{2} x -10 b^{5} c x +640 a^{3} c^{3}-576 a^{2} b^{2} c^{2}-70 a \,b^{4} c +5 b^{6}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{4}}{2310 c^{2} \sqrt {d \left (2 c x +b \right )}}+\frac {\left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{462 c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(642\)
default	\(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{3} \left (-1536 a^{3} b \,c^{5} x^{2}+42688 a^{2} b^{2} c^{5} x^{3}+87808 a b \,c^{7} x^{6}+123328 a \,b^{2} c^{6} x^{5}+41920 a^{2} b \,c^{6} x^{4}+88800 a \,b^{3} c^{5} x^{4}+33728 a \,b^{4} c^{4} x^{3}+22112 a^{2} b^{3} c^{4} x^{2}+5696 a \,b^{5} c^{3} x^{2}+1792 a^{3} b^{2} c^{4} x +4856 a^{2} b^{4} c^{3} x +140 a \,b^{6} c^{2} x +9856 c^{9} x^{9}-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 +13988 b^{5} c^{4} x^{4}+16768 a^{2} c^{7} x^{5}+81536 b^{3} c^{6} x^{6}+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}+44352 b \,c^{8} x^{8}+25088 a \,c^{8} x^{7}+82432 b^{2} c^{7} x^{7}+45736 b^{4} c^{5} x^{5}-1024 a^{3} c^{6} x^{3}+1852 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}+1152 a^{3} b^{3} c^{3}-10 a \,b^{7} c +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 )}{4620 c^{3} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\)	\(1057\)
elliptic	\(\text {Expression too large to display}\)	\(5816\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4620*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(-1536*a^3*b*c^5*x^2+42688*a^2*b^2*c^5*x^3+87808*a*b*c^7*x^
6+123328*a*b^2*c^6*x^5+41920*a^2*b*c^6*x^4+88800*a*b^3*c^5*x^4+33728*a*b^4*c^4*x^3+22112*a^2*b^3*c^4*x^2+5696*
a*b^5*c^3*x^2+1792*a^3*b^2*c^4*x+4856*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)*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)*a*b^6
*c+9856*c^9*x^9+13988*b^5*c^4*x^4+16768*a^2*c^7*x^5+81536*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)*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)*b
^8+44352*b*c^8*x^8+25088*a*c^8*x^7+82432*b^2*c^7*x^7+45736*b^4*c^5*x^5-1024*a^3*c^6*x^3+1852*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+1152*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^3/(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)^(7/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.29, size = 292, normalized size = 1.07 \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^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (2464 \, c^{8} d^{3} x^{6} + 7392 \, b c^{7} d^{3} x^{5} + 224 \, {\left (37 \, b^{2} c^{6} + 17 \, a c^{7}\right )} d^{3} x^{4} + 224 \, {\left (19 \, b^{3} c^{5} + 34 \, a b c^{6}\right )} d^{3} x^{3} + 6 \, {\left (151 \, b^{4} c^{4} + 920 \, a b^{2} c^{5} + 64 \, a^{2} c^{6}\right )} d^{3} x^{2} + 2 \, {\left (5 \, b^{5} c^{3} + 856 \, a b^{3} c^{4} + 192 \, a^{2} b c^{5}\right )} d^{3} x - {\left (5 \, b^{6} c^{2} - 70 \, a b^{4} c^{3} - 576 \, a^{2} b^{2} c^{4} + 640 \, a^{3} c^{5}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{4620 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

1/4620*(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^3*weierstr
assPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(2464*c^8*d^3*x^6 + 7392*b*c^7*d^3*x^5 + 224*(37*b^2*
c^6 + 17*a*c^7)*d^3*x^4 + 224*(19*b^3*c^5 + 34*a*b*c^6)*d^3*x^3 + 6*(151*b^4*c^4 + 920*a*b^2*c^5 + 64*a^2*c^6)
*d^3*x^2 + 2*(5*b^5*c^3 + 856*a*b^3*c^4 + 192*a^2*b*c^5)*d^3*x - (5*b^6*c^2 - 70*a*b^4*c^3 - 576*a^2*b^2*c^4 +
640*a^3*c^5)*d^3)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^4

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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 {7}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral((d*(b + 2*c*x))**(7/2)*(a + b*x + c*x**2)**(3/2), 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)^(7/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(3/2), 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 )}^{7/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2),x)

[Out]

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2), x)

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