3.14.27 \(\int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx\) [1327]

Optimal. Leaf size=227 \[ -\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{231 c^2 \sqrt {a+b x+c x^2}} \]

[Out]

-2/77*(-4*a*c+b^2)*d*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c+1/11*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(1/2)/c/
d-10/231*(-4*a*c+b^2)^2*d^3*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c-5/231*(-4*a*c+b^2)^(13/4)*d^(7/2)*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^2/(c*x^2+b*x+a)^
(1/2)

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Rubi [A]
time = 0.14, 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 {5 d^{7/2} \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 )}{231 c^2 \sqrt {a+b x+c x^2}}-\frac {10 d^3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c}-\frac {2 d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{77 c}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{11 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2],x]

[Out]

(-10*(b^2 - 4*a*c)^2*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(231*c) - (2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*
x)^(5/2)*Sqrt[a + b*x + c*x^2])/(77*c) + ((b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(11*c*d) - (5*(b^2 - 4*
a*c)^(13/4)*d^(7/2)*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])/(231*c^2*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)^{7/2} \sqrt {a+b x+c x^2} \, dx &=\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{22 c}\\ &=-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^2 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{154 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{462 c}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^4 \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}{462 c \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3 d^3 \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 )}{231 c^2 \sqrt {a+b x+c x^2}}\\ &=-\frac {10 \left (b^2-4 a c\right )^2 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c}-\frac {2 \left (b^2-4 a c\right ) d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{77 c}+\frac {(b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} d^{7/2} \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 )}{231 c^2 \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 = 10.29, size = 155, normalized size = 0.68 \begin {gather*} \frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (7 (b+2 c x)^2 (a+x (b+c x))-10 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))+\frac {\left (b^2-4 a c\right ) \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{77 (b+2 c x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(7/2)*Sqrt[a + b*x + c*x^2],x]

[Out]

(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(7*(b + 2*c*x)^2*(a + x*(b + c*x)) - 10*(a - b^2/(4*c))*c*(2*(a
+ x*(b + c*x)) + ((b^2 - 4*a*c)*Hypergeometric2F1[-1/2, 1/4, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(4*c*Sqrt[(c*
(a + x*(b + c*x)))/(-b^2 + 4*a*c)]))))/(77*(b + 2*c*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(797\) vs. \(2(193)=386\).
time = 0.99, size = 798, normalized size = 3.52

method	result	size
risch	\(-\frac {\left (-336 c^{4} x^{4}-672 b \,c^{3} x^{3}-96 x^{2} c^{3} a -480 b^{2} c^{2} x^{2}-96 x a b \,c^{2}-144 b^{3} c x +160 a^{2} c^{2}-104 a c \,b^{2}-5 b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{4}}{231 c \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{231 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a c d x +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(555\)
default	\(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{3} \left (1344 c^{7} x^{7}+4704 b \,c^{6} x^{6}+1728 a \,c^{6} x^{5}+6624 b^{2} c^{5} x^{5}+320 \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}-240 \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}+60 \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 -5 \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}+4320 a b \,c^{5} x^{4}+4800 b^{3} c^{4} x^{4}-256 a^{2} c^{5} x^{3}+4448 a \,b^{2} c^{4} x^{3}+1844 b^{4} c^{3} x^{3}-384 a^{2} b \,c^{4} x^{2}+2352 a \,b^{3} c^{3} x^{2}+318 b^{5} c^{2} x^{2}-640 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x +516 a \,b^{4} c^{2} x +10 b^{6} c x -320 a^{3} b \,c^{3}+208 a^{2} b^{3} c^{2}+10 a \,b^{5} c \right )}{462 c^{2} \left (2 c^{2} x^{3}+3 c \,x^{2} b +2 a c x +b^{2} x +a b \right )}\)	\(798\)
elliptic	\(\text {Expression too large to display}\)	\(2307\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/462*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(1344*c^7*x^7+4704*b*c^6*x^6+1728*a*c^6*x^5+6624*b^2*c^5*x^5
+320*((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^3-240*((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+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)*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-5*((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+4320*a*b*c^5*x^4+4800*b^3*
c^4*x^4-256*a^2*c^5*x^3+4448*a*b^2*c^4*x^3+1844*b^4*c^3*x^3-384*a^2*b*c^4*x^2+2352*a*b^3*c^3*x^2+318*b^5*c^2*x
^2-640*a^3*c^4*x+288*a^2*b^2*c^3*x+516*a*b^4*c^2*x+10*b^6*c*x-320*a^3*b*c^3+208*a^2*b^3*c^2+10*a*b^5*c)/c^2/(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)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.80, size = 199, normalized size = 0.88 \begin {gather*} -\frac {5 \, \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^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (336 \, c^{6} d^{3} x^{4} + 672 \, b c^{5} d^{3} x^{3} + 96 \, {\left (5 \, b^{2} c^{4} + a c^{5}\right )} d^{3} x^{2} + 48 \, {\left (3 \, b^{3} c^{3} + 2 \, a b c^{4}\right )} d^{3} x + {\left (5 \, b^{4} c^{2} + 104 \, a b^{2} c^{3} - 160 \, a^{2} c^{4}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{462 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

-1/462*(5*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^3*weierstrassPInverse((b^2 -
4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(336*c^6*d^3*x^4 + 672*b*c^5*d^3*x^3 + 96*(5*b^2*c^4 + a*c^5)*d^3*x^2 +
48*(3*b^3*c^3 + 2*a*b*c^4)*d^3*x + (5*b^4*c^2 + 104*a*b^2*c^3 - 160*a^2*c^4)*d^3)*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 {7}{2}} \sqrt {a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(7/2)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d*(b + 2*c*x))**(7/2)*sqrt(a + b*x + c*x**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)^(7/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*sqrt(c*x^2 + b*x + a), 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 )}^{7/2}\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2),x)

[Out]

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(1/2), x)

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