3.14.44 \(\int (b d+2 c d x)^{5/2} (a+b x+c x^2)^{3/2} \, dx\) [1344]

Optimal. Leaf size=326 \[ \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{130 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{130 c^3 \sqrt {a+b x+c x^2}} \]

[Out]

1/13*(2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(3/2)/c/d+1/195*(-4*a*c+b^2)^2*d*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2
)/c^2-1/78*(-4*a*c+b^2)*(2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2)/c^2/d+1/130*(-4*a*c+b^2)^(15/4)*d^(5/2)*Ellipt
icE((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)-1/130*(-4*a*c+b^2)^(15/4)*d^(5/2)*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 = 0.22, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {699, 706, 705, 704, 313, 227, 1213, 435} \begin {gather*} -\frac {d^{5/2} \left (b^2-4 a c\right )^{15/4} \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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(195*c^2) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(7/
2)*Sqrt[a + b*x + c*x^2])/(78*c^2*d) + ((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))/(13*c*d) + ((b^2 - 4*a*
c)^(15/4)*d^(5/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[b*d + 2*c*d*x]/((b^2 - 4*
a*c)^(1/4)*Sqrt[d])], -1])/(130*c^3*Sqrt[a + b*x + c*x^2]) - ((b^2 - 4*a*c)^(15/4)*d^(5/2)*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])/(130*c^3*S
qrt[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 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && 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 704

Int[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[x^2/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])

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps

\begin {align*} \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2} \, dx}{26 c}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx}{156 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{260 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {\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}{260 c^2 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{130 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^{7/2} d^2 \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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{7/2} d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{130 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{7/2} d^2 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{130 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{130 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \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 )}{130 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 = 10.10, size = 117, normalized size = 0.36 \begin {gather*} \frac {2}{13} d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^2-\frac {\left (b^2-4 a c\right )^2 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^2 - ((b^2 - 4*a*c)^2*Hypergeometric2F1[-
3/2, 3/4, 7/4, (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)])))/13

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(937\) vs. \(2(276)=552\).
time = 0.80, size = 938, normalized size = 2.88

method	result	size
default	\(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{2} \left (-7680 a b \,c^{6} x^{5}-8672 a \,b^{2} c^{5} x^{4}-3712 a^{2} b \,c^{5} x^{3}-4544 a \,b^{3} c^{4} x^{3}-2592 a^{2} b^{2} c^{4} x^{2}-1056 a \,b^{4} c^{3} x^{2}-256 a^{3} b \,c^{4} x -736 a^{2} b^{3} c^{3} x -64 a \,b^{5} c^{2} x +10 b^{6} c^{2} x^{2}-6080 b^{2} c^{6} x^{6}-4800 b^{3} c^{5} x^{5}-64 a^{3} b^{2} c^{3}-68 a^{2} b^{4} c^{2}-960 c^{8} x^{8}+6 a \,b^{6} c -1916 b^{4} c^{4} x^{4}-312 b^{5} c^{3} x^{3}+6 b^{7} c x -3840 b \,c^{7} x^{7}-2560 a \,c^{7} x^{6}-1856 a^{2} c^{6} x^{4}-256 a^{3} c^{5} x^{2}-768 \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}}}}\, \EllipticE \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 ) a^{3} b^{2} c^{3}+288 \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}}}}\, \EllipticE \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 ) a^{2} b^{4} c^{2}-48 \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}}}}\, \EllipticE \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 ) a \,b^{6} c +768 \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}}}}\, \EllipticE \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 ) a^{4} c^{4}+3 \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}}}}\, \EllipticE \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 ) b^{8}\right )}{780 c^{3} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\)	\(938\)
risch	\(\text {Expression too large to display}\)	\(2234\)
elliptic	\(\text {Expression too large to display}\)	\(3580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/780*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(-7680*a*b*c^6*x^5-8672*a*b^2*c^5*x^4-3712*a^2*b*c^5*x^3-45
44*a*b^3*c^4*x^3-2592*a^2*b^2*c^4*x^2-1056*a*b^4*c^3*x^2-256*a^3*b*c^4*x-736*a^2*b^3*c^3*x-64*a*b^5*c^2*x+10*b
^6*c^2*x^2-6080*b^2*c^6*x^6-4800*b^3*c^5*x^5+768*((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)*EllipticE(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))*a^4*c^4+3*((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)*EllipticE(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))*b^8-64*a^3*b^2*
c^3-68*a^2*b^4*c^2-960*c^8*x^8-768*((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)*EllipticE(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))*a^3*b^2*c^3+288*((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)*
EllipticE(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))*a^2*b^4*c^2-48*((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)*EllipticE(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))*a*b^6*c+6*a*b^6*c-1916*b^4*c^4*x^4-312*b^5*c^3*x^3+6*b^7*c*x-3840*b*c^7*x^7-2560*a*c^7*x^6-1856*a^2*
c^6*x^4-256*a^3*c^5*x^2)/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)^(5/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(5/2)*(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 = 0.56, size = 246, normalized size = 0.75 \begin {gather*} -\frac {3 \, \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^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (240 \, c^{6} d^{2} x^{5} + 600 \, b c^{5} d^{2} x^{4} + 100 \, {\left (5 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{2} x^{3} + 150 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{2} x^{2} + 4 \, {\left (b^{4} c^{2} + 67 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 34 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{390 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/390*(3*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^2*weierstrassZeta((b^2 - 4*a*
c)/c^2, 0, weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) - (240*c^6*d^2*x^5 + 600*b*c^5*d^2*x^
4 + 100*(5*b^2*c^4 + 4*a*c^5)*d^2*x^3 + 150*(b^3*c^3 + 4*a*b*c^4)*d^2*x^2 + 4*(b^4*c^2 + 67*a*b^2*c^3 + 16*a^2
*c^4)*d^2*x - (3*b^5*c - 34*a*b^3*c^2 - 32*a^2*b*c^3)*d^2)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^3

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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 {5}{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)**(5/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

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

[Out]

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

[Out]

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

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