3.14.38 \(\int (b d+2 c d x)^{3/2} (a+b x+c x^2)^{3/2} \, dx\) [1338]

Optimal. Leaf size=227 \[ \frac {\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}+\frac {\left (b^2-4 a c\right )^{13/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{154 c^3 \sqrt {a+b x+c x^2}} \]

[Out]

1/11*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(3/2)/c/d-3/154*(-4*a*c+b^2)*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c^
2/d+1/77*(-4*a*c+b^2)^2*d*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2+1/154*(-4*a*c+b^2)^(13/4)*d^(3/2)*Ellipt
icF((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 = 0.13, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 6, 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^{3/2} \left (b^2-4 a c\right )^{13/4} \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 )}{154 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} \sqrt {b d+2 c d x}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{154 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{5/2}}{11 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(77*c^2) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5/2
)*Sqrt[a + b*x + c*x^2])/(154*c^2*d) + ((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^(3/2))/(11*c*d) + ((b^2 - 4*a*
c)^(13/4)*d^(3/2)*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])/(154*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)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2} \, dx}{22 c}\\ &=-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}+\frac {\left (3 \left (b^2-4 a c\right )^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 )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{308 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 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 {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}{308 c^2 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 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 {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 )}{154 c^3 \sqrt {a+b x+c x^2}}\\ &=\frac {\left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{77 c^2}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{154 c^2 d}+\frac {(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}}{11 c d}+\frac {\left (b^2-4 a c\right )^{13/4} d^{3/2} \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 )}{154 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 = 10.07, size = 117, normalized size = 0.52 \begin {gather*} \frac {2}{11} d \sqrt {d (b+2 c x)} \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 {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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*(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)])))/11

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(795\) vs. \(2(193)=386\).
time = 0.78, size = 796, normalized size = 3.51

method	result	size
risch	\(\frac {\left (56 c^{4} x^{4}+112 b \,c^{3} x^{3}+104 x^{2} c^{3} a +58 b^{2} c^{2} x^{2}+104 x a b \,c^{2}+2 b^{3} c x +32 a^{2} c^{2}+10 a c \,b^{2}-b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{2}}{154 c^{2} \sqrt {d \left (2 c x +b \right )}}-\frac {\left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{154 c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(555\)
default	\(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (-224 c^{7} x^{7}-784 b \,c^{6} x^{6}-640 a \,c^{6} x^{5}-1016 b^{2} c^{5} x^{5}+64 \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} c^{3}-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}}}}\, \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^{2} c^{2}+12 \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^{4} c -\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^{6}-1600 a b \,c^{5} x^{4}-580 b^{3} c^{4} x^{4}-544 a^{2} c^{5} x^{3}-1328 a \,b^{2} c^{4} x^{3}-124 b^{4} c^{3} x^{3}-816 a^{2} b \,c^{4} x^{2}-392 a \,b^{3} c^{3} x^{2}+2 b^{5} c^{2} x^{2}-128 a^{3} c^{4} x -312 a^{2} b^{2} c^{3} x -20 a \,b^{4} c^{2} x +2 b^{6} c x -64 a^{3} b \,c^{3}-20 a^{2} b^{3} c^{2}+2 a \,b^{5} c \right )}{308 c^{3} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\)	\(796\)
elliptic	\(\text {Expression too large to display}\)	\(2281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/308*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d*(-224*c^7*x^7-784*b*c^6*x^6-640*a*c^6*x^5-1016*b^2*c^5*x^5+64
*((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*c^3-48*((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)*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^2*c^2+12*((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*b^4*c-((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^6-1600*a*b*c^5*x^4-580*b^3*c^4*x^4-
544*a^2*c^5*x^3-1328*a*b^2*c^4*x^3-124*b^4*c^3*x^3-816*a^2*b*c^4*x^2-392*a*b^3*c^3*x^2+2*b^5*c^2*x^2-128*a^3*c
^4*x-312*a^2*b^2*c^3*x-20*a*b^4*c^2*x+2*b^6*c*x-64*a^3*b*c^3-20*a^2*b^3*c^2+2*a*b^5*c)/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)^(3/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(3/2)*(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 = 0.52, size = 186, normalized size = 0.82 \begin {gather*} \frac {\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 {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (56 \, c^{6} d x^{4} + 112 \, b c^{5} d x^{3} + 2 \, {\left (29 \, b^{2} c^{4} + 52 \, a c^{5}\right )} d x^{2} + 2 \, {\left (b^{3} c^{3} + 52 \, a b c^{4}\right )} d x - {\left (b^{4} c^{2} - 10 \, a b^{2} c^{3} - 32 \, a^{2} c^{4}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{308 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/308*(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*weierstrassPInverse((b^2 - 4*a*c
)/c^2, 0, 1/2*(2*c*x + b)/c) + 2*(56*c^6*d*x^4 + 112*b*c^5*d*x^3 + 2*(29*b^2*c^4 + 52*a*c^5)*d*x^2 + 2*(b^3*c^
3 + 52*a*b*c^4)*d*x - (b^4*c^2 - 10*a*b^2*c^3 - 32*a^2*c^4)*d)*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 {3}{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)**(3/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

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

[Out]

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

[Out]

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

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