\(\int (b d+2 c d x)^{5/2} (a+b x+c x^2)^{5/2} \, dx\) [1357]

Optimal result

Integrand size = 28, antiderivative size = 373 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{442 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2}}{17 c d}-\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{19/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{884 c^4 \sqrt {a+b x+c x^2}} \]

[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)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {699, 706, 705, 704, 313, 227, 1213, 435} \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d^{5/2} \left (b^2-4 a c\right )^{19/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{884 c^4 \sqrt {a+b x+c x^2}}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{1326 c^3}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{2652 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{442 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{7/2}}{17 c d} \]

[In]

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

[Out]

-1/1326*((b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/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 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*} \text {integral}& = \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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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 ) \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 )}{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*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.31 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{17} d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^3+\frac {\left (b^2-4 a c\right )^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1189\) vs. \(2(317)=634\).

Time = 4.49 (sec) , antiderivative size = 1190, normalized size of antiderivative = 3.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1190\)
risch	\(\text {Expression too large to display}\)	\(2710\)
elliptic	\(\text {Expression too large to display}\)	\(9950\)

[In]

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

[Out]

-1/5304*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^2*(-34168*b^4*c^6*x^6-1516*b^6*c^4*x^4-51456*b^2*c^8*x^8-249
60*b*c^9*x^9-11112*b^5*c^5*x^5-6*b^9*c*x-56064*b^3*c^7*x^7-3840*((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)*Ellip
ticE(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*b^2*c^4+1920*((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^4*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)*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^6*c^2+60*((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^8*c-4992*c^10*x^10-75264*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^4-10
24*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-18816*a*c^9*x^8-25216*a^2*c^8*x^6-12416*a^3
*c^7*x^4-1024*a^4*c^6*x^2-24832*a^3*b*c^6*x^3-44416*a^2*b^3*c^5*x^3-6064*a*b^5*c^4*x^3-3*((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^1
0-256*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-10*b^8*c^2*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+3072*((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^5*c^5)/c^4/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a*b)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.93 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {3 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (1248 \, c^{8} d^{2} x^{7} + 4368 \, b c^{7} d^{2} x^{6} + 72 \, {\left (79 \, b^{2} c^{6} + 48 \, a c^{7}\right )} d^{2} x^{5} + 60 \, {\left (55 \, b^{3} c^{5} + 144 \, a b c^{6}\right )} d^{2} x^{4} + 4 \, {\left (187 \, b^{4} c^{4} + 1804 \, a b^{2} c^{5} + 712 \, a^{2} c^{6}\right )} d^{2} x^{3} + 6 \, {\left (b^{5} c^{3} + 364 \, a b^{3} c^{4} + 712 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 4 \, {\left (b^{6} c^{2} - 15 \, a b^{4} c^{3} - 486 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x + {\left (3 \, b^{7} c - 46 \, a b^{5} c^{2} + 260 \, a^{2} b^{3} c^{3} + 128 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{2652 \, c^{4}} \]

[In]

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

[Out]

1/2652*(3*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^2*weierstr
assZeta((b^2 - 4*a*c)/c^2, 0, weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c)) + (1248*c^8*d^2*x^
7 + 4368*b*c^7*d^2*x^6 + 72*(79*b^2*c^6 + 48*a*c^7)*d^2*x^5 + 60*(55*b^3*c^5 + 144*a*b*c^6)*d^2*x^4 + 4*(187*b
^4*c^4 + 1804*a*b^2*c^5 + 712*a^2*c^6)*d^2*x^3 + 6*(b^5*c^3 + 364*a*b^3*c^4 + 712*a^2*b*c^5)*d^2*x^2 - 4*(b^6*
c^2 - 15*a*b^4*c^3 - 486*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*x + (3*b^7*c - 46*a*b^5*c^2 + 260*a^2*b^3*c^3 + 128*a^3
*b*c^4)*d^2)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^4

Sympy [F]

\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]

[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)

Maxima [F]

\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]

[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)

Giac [F]

\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]

[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)