3.1357 \(\int (b d+2 c d x)^{5/2} (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=373 \[ \frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]

[Out]

-5/442*(-4*a*c+b^2)*(2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(3/2)/c^2/d+1/17*(2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(5/2)
/c/d-1/1326*(-4*a*c+b^2)^3*d*(2*c*d*x+b*d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^3+5/2652*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(
7/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-1/884*(-4*a*c+b^2)^(19/4)*d^(5/2)*EllipticE((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)+1/884*(-4*a*c+b^2)^(19/4)*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^4/(c*x^2+
b*x+a)^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {685, 692, 691, 690, 307, 221, 1199, 424} \[ \frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(1326*c^3) + (5*(b^2 - 4*a*c)^2*(b*d + 2*c*d*
x)^(7/2)*Sqrt[a + b*x + c*x^2])/(2652*c^3*d) - (5*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(3/2))
/(442*c^2*d) + ((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2))/(17*c*d) - ((b^2 - 4*a*c)^(19/4)*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
])/(884*c^4*Sqrt[a + b*x + c*x^2]) + ((b^2 - 4*a*c)^(19/4)*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])/(884*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 692

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 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 (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx}{34 c}\\ &=-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2} \, dx}{884 c^2}\\ &=\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx}{5304 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (\left (b^2-4 a c\right )^4 d^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{1768 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 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 {\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}{1768 c^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 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 ) \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 )}{884 c^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (\left (b^2-4 a c\right )^{9/2} d^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 )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{9/2} d^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 )}{884 c^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}+\frac {\left (b^2-4 a c\right )^{19/4} 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 )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{9/2} d^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 )}{884 c^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (b^2-4 a c\right )^{19/4} 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 )}{884 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{19/4} 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 )}{884 c^4 \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.18, size = 117, normalized size = 0.31 \[ \frac {2}{17} d \sqrt {a+x (b+c x)} (d (b+2 c x))^{3/2} \left (\frac {\left (b^2-4 a c\right )^3 \, _2F_1\left (-\frac {5}{2},\frac {3}{4};\frac {7}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}}+2 (a+x (b+c x))^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(d*(b + 2*c*x))^(3/2)*Sqrt[a + x*(b + c*x)]*(2*(a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3*Hypergeometric2F1[-
5/2, 3/4, 7/4, (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)])))/17

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (4 \, c^{4} d^{2} x^{6} + 12 \, b c^{3} d^{2} x^{5} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{4} + a^{2} b^{2} d^{2} + 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{3} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((4*c^4*d^2*x^6 + 12*b*c^3*d^2*x^5 + (13*b^2*c^2 + 8*a*c^3)*d^2*x^4 + a^2*b^2*d^2 + 2*(3*b^3*c + 8*a*b
*c^2)*d^2*x^3 + (b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*x^2 + 2*(a*b^3 + 2*a^2*b*c)*d^2*x)*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 \[ \int {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 1190, normalized size = 3.19 \[ -\frac {\sqrt {\left (2 c x +b \right ) d}\, \sqrt {c \,x^{2}+b x +a}\, \left (-4992 c^{10} x^{10}-24960 b \,c^{9} x^{9}-18816 a \,c^{9} x^{8}-51456 b^{2} c^{8} x^{8}-75264 a b \,c^{8} x^{7}-56064 b^{3} c^{7} x^{7}-25216 a^{2} c^{8} x^{6}-119104 a \,b^{2} c^{7} x^{6}-34168 b^{4} c^{6} x^{6}-75648 a^{2} b \,c^{7} x^{5}-93888 a \,b^{3} c^{6} x^{5}-11112 b^{5} c^{5} x^{5}-12416 a^{3} c^{7} x^{4}-85248 a^{2} b^{2} c^{6} x^{4}-37368 a \,b^{4} c^{5} x^{4}-1516 b^{6} c^{4} x^{4}-24832 a^{3} b \,c^{6} x^{3}-44416 a^{2} b^{3} c^{5} x^{3}-6064 a \,b^{5} c^{4} x^{3}-1024 a^{4} c^{6} x^{2}-17600 a^{3} b^{2} c^{5} x^{2}-10056 a^{2} b^{4} c^{4} x^{2}+160 a \,b^{6} c^{3} x^{2}-10 b^{8} c^{2} x^{2}+3072 \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^{5} c^{5} \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 )-3840 \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} b^{2} 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 )-1024 a^{4} b \,c^{5} x +1920 \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^{4} 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 )-5184 a^{3} b^{3} c^{4} x -480 \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^{6} 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 )-456 a^{2} b^{5} c^{3} x +60 \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^{8} 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 )+88 a \,b^{7} 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^{10} \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^{9} c x -256 a^{4} b^{2} c^{4}-520 a^{3} b^{4} c^{3}+92 a^{2} b^{6} c^{2}-6 a \,b^{8} c \right ) d^{2}}{5304 \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)^(5/2)*(c*x^2+b*x+a)^(5/2),x)

[Out]

-1/5304*((2*c*x+b)*d)^(1/2)*(c*x^2+b*x+a)^(1/2)*d^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^10-4992*c^10*x^10-256*a^4*b^2*c^4-
520*a^3*b^4*c^3+92*a^2*b^6*c^2-6*a*b^8*c-1024*a^4*c^6*x^2+160*a*b^6*c^3*x^2-1024*a^4*b*c^5*x-5184*a^3*b^3*c^4*
x-456*a^2*b^5*c^3*x+88*a*b^7*c^2*x-75264*a*b*c^8*x^7-119104*a*b^2*c^7*x^6-75648*a^2*b*c^7*x^5-93888*a*b^3*c^6*
x^5-85248*a^2*b^2*c^6*x^4-37368*a*b^4*c^5*x^4+3072*((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^5*c^5-24832*a^3*b*c^6*x^3-44416*a^2*b^3
*c^5*x^3-6064*a*b^5*c^4*x^3-17600*a^3*b^2*c^5*x^2-10056*a^2*b^4*c^4*x^2-6*b^9*c*x-11112*b^5*c^5*x^5-56064*b^3*
c^7*x^7-34168*b^4*c^6*x^6-51456*b^2*c^8*x^8-24960*b*c^9*x^9-10*b^8*c^2*x^2-1516*b^6*c^4*x^4-18816*a*c^9*x^8-25
216*a^2*c^8*x^6-12416*a^3*c^7*x^4-3840*((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*b^2*c^4+1920*((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^4*c^3-480*((
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^6*c^2+60*((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^8*c)/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 \[ \int {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)**(5/2),x)

[Out]

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

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