3.1358 \(\int \sqrt {b d+2 c d x} (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=328 \[ \frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{3/2}}{13 c d} \]

[Out]

-5/234*(-4*a*c+b^2)*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(3/2)/c^2/d+1/13*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(5/2)
/c/d+1/156*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-1/156*(-4*a*c+b^2)^(15/4)*EllipticE((2
*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*d^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/(c*x^2+b*x+a
)^(1/2)+1/156*(-4*a*c+b^2)^(15/4)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b^2)^(1/4)/d^(1/2),I)*d^(1/2)*(-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 = 0.30, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {685, 691, 690, 307, 221, 1199, 424} \[ \frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{3/2}}{13 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(156*c^3*d) - (5*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(
3/2)*(a + b*x + c*x^2)^(3/2))/(234*c^2*d) + ((b*d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^(5/2))/(13*c*d) - ((b^2 -
4*a*c)^(15/4)*Sqrt[d]*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])/(156*c^4*Sqrt[a + b*x + c*x^2]) + ((b^2 - 4*a*c)^(15/4)*Sqrt[d]*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])/(156*
c^4*Sqrt[a + b*x + c*x^2])

Rule 221

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 307

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 424

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

Rule 685

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 690

Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(4*Sqrt[-(c/(b^2 - 4*a*
c))])/e, 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 691

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 1199

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 \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx}{26 c}\\ &=-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx}{156 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^3 \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{312 c^3}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^3 \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}{312 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^3 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {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 )}{156 c^4 d \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {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 )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {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 )}{156 c^4 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} \sqrt {d} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {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 )}{156 c^4 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^{15/4} \sqrt {d} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{15/4} \sqrt {d} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 101, normalized size = 0.31 \[ \frac {\left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)} (d (b+2 c x))^{3/2} \, _2F_1\left (-\frac {5}{2},\frac {3}{4};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{96 c^3 d \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)^2*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-5/2, 3/4, 7/4, (b + 2*c*x)^2/(
b^2 - 4*a*c)])/(96*c^3*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a),
x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.08, size = 924, normalized size = 2.82 \[ \frac {\sqrt {\left (2 c x +b \right ) d}\, \sqrt {c \,x^{2}+b x +a}\, \left (288 c^{8} x^{8}+1152 b \,c^{7} x^{7}+1184 a \,c^{7} x^{6}+1720 b^{2} c^{6} x^{6}+3552 a b \,c^{6} x^{5}+1128 b^{3} c^{5} x^{5}+1888 a^{2} c^{6} x^{4}+3496 a \,b^{2} c^{5} x^{4}+268 b^{4} c^{4} x^{4}+3776 a^{2} b \,c^{5} x^{3}+1072 a \,b^{3} c^{4} x^{3}+992 a^{3} c^{5} x^{2}+2088 a^{2} b^{2} c^{4} x^{2}-120 a \,b^{4} c^{3} x^{2}+10 b^{6} c^{2} x^{2}+768 \sqrt {\frac {2 c x +b +\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 {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a^{4} c^{4} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-768 \sqrt {\frac {2 c x +b +\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 {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a^{3} b^{2} c^{3} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+992 a^{3} b \,c^{4} x +288 \sqrt {\frac {2 c x +b +\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 {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a^{2} b^{4} c^{2} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+200 a^{2} b^{3} c^{3} x -48 \sqrt {\frac {2 c x +b +\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 {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, a \,b^{6} c \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-64 a \,b^{5} c^{2} x +3 \sqrt {\frac {2 c x +b +\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 {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, b^{8} \EllipticE \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )+6 b^{7} c x +248 a^{3} b^{2} c^{3}-68 a^{2} b^{4} c^{2}+6 a \,b^{6} c \right )}{936 \left (2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(5/2),x)

[Out]

1/936*((2*c*x+b)*d)^(1/2)*(c*x^2+b*x+a)^(1/2)*(3*((2*c*x+b+(-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)*((-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((2*c*
x+b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2))*b^8+768*((2*c*x+b+(-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)*((-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^
(1/2)*EllipticE(1/2*((2*c*x+b+(-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+248*a^3*b
^2*c^3-68*a^2*b^4*c^2+6*a*b^6*c+1128*b^3*c^5*x^5+10*b^6*c^2*x^2+6*b^7*c*x+1720*b^2*c^6*x^6+1152*b*c^7*x^7+268*
b^4*c^4*x^4+1184*a*c^7*x^6+1888*a^2*c^6*x^4+992*a^3*c^5*x^2+288*c^8*x^8-64*a*b^5*c^2*x+3552*a*b*c^6*x^5+3496*a
*b^2*c^5*x^4+3776*a^2*b*c^5*x^3+1072*a*b^3*c^4*x^3+2088*a^2*b^2*c^4*x^2-120*a*b^4*c^3*x^2+992*a^3*b*c^4*x+200*
a^2*b^3*c^3*x-48*((2*c*x+b+(-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)
*((-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((2*c*x+b+(-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-768*((2*c*x+b+(-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)*((-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticE(1/2*((2*c*x+
b+(-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*((2*c*x+b+(-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)*((-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(
1/2))^(1/2)*EllipticE(1/2*((2*c*x+b+(-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
)/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 \[ \int \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {b\,d+2\,c\,d\,x}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(1/2)*(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^(1/2)*(a + b*x + c*x^2)^(5/2), x)

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sympy [A]  time = 18.21, size = 539, normalized size = 1.64 \[ \frac {a^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{4 c d \Gamma \left (\frac {7}{4}\right )} - \frac {a b^{2} \left (b d + 2 c d x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{8 c^{2} d \Gamma \left (\frac {7}{4}\right )} + \frac {a \left (b d + 2 c d x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{8 c^{2} d^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {b^{4} \left (b d + 2 c d x\right )^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{64 c^{3} d \Gamma \left (\frac {7}{4}\right )} - \frac {b^{2} \left (b d + 2 c d x\right )^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{32 c^{3} d^{3} \Gamma \left (\frac {11}{4}\right )} + \frac {\left (b d + 2 c d x\right )^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {\left (b d + 2 c d x\right )^{2} e^{i \pi }}{4 c d^{2} \operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}} \right )} \sqrt {\operatorname {polar\_lift}{\left (a - \frac {b^{2}}{4 c} \right )}}}{64 c^{3} d^{5} \Gamma \left (\frac {15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a)**(5/2),x)

[Out]

a**2*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2
*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(4*c*d*gamma(7/4)) - a*b**2*(b*d + 2*c*d*x)**(3
/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*
c))))*sqrt(polar_lift(a - b**2/(4*c)))/(8*c**2*d*gamma(7/4)) + a*(b*d + 2*c*d*x)**(7/2)*gamma(7/4)*hyper((-1/2
, 7/4), (11/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a -
b**2/(4*c)))/(8*c**2*d**3*gamma(11/4)) + b**4*(b*d + 2*c*d*x)**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), (b
*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(64*c
**3*d*gamma(7/4)) - b**2*(b*d + 2*c*d*x)**(7/2)*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), (b*d + 2*c*d*x)**2*exp_
polar(I*pi)/(4*c*d**2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(32*c**3*d**3*gamma(11/4))
+ (b*d + 2*c*d*x)**(11/2)*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), (b*d + 2*c*d*x)**2*exp_polar(I*pi)/(4*c*d*
*2*polar_lift(a - b**2/(4*c))))*sqrt(polar_lift(a - b**2/(4*c)))/(64*c**3*d**5*gamma(15/4))

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