3.14.49 \(\int (b d+2 c d x)^{7/2} (a+b x+c x^2)^{5/2} \, dx\) [1349]

Optimal. Leaf size=321 \[ -\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right )^{21/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{17556 c^4 \sqrt {a+b x+c x^2}} \]

[Out]

-1/114*(-4*a*c+b^2)*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(3/2)/c^2/d+1/19*(2*c*d*x+b*d)^(9/2)*(c*x^2+b*x+a)^(5/2)
/c/d-1/2926*(-4*a*c+b^2)^3*d*(2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c^3+1/836*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(9
/2)*(c*x^2+b*x+a)^(1/2)/c^3/d-5/8778*(-4*a*c+b^2)^4*d^3*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3-5/17556*(-
4*a*c+b^2)^(21/4)*d^(7/2)*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 = 0.21, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 8, 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 )^{21/4} \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 )}{17556 c^4 \sqrt {a+b x+c x^2}}-\frac {5 d^3 \left (b^2-4 a c\right )^4 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{8778 c^3}-\frac {d \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{114 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{9/2}}{19 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(-5*(b^2 - 4*a*c)^4*d^3*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(8778*c^3) - ((b^2 - 4*a*c)^3*d*(b*d + 2*c*
d*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(2926*c^3) + ((b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(9/2)*Sqrt[a + b*x + c*x^2])/(
836*c^3*d) - ((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(9/2)*(a + b*x + c*x^2)^(3/2))/(114*c^2*d) + ((b*d + 2*c*d*x)^(9/2
)*(a + b*x + c*x^2)^(5/2))/(19*c*d) - (5*(b^2 - 4*a*c)^(21/4)*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])/(17556*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 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} \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx}{38 c}\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}+\frac {\left (b^2-4 a c\right )^2 \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx}{76 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (b^2-4 a c\right )^3 \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{1672 c^3}\\ &=-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (5 \left (b^2-4 a c\right )^4 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{11704 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (5 \left (b^2-4 a c\right )^5 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{35112 c^3}\\ &=-\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (5 \left (b^2-4 a c\right )^5 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}{35112 c^3 \sqrt {a+b x+c x^2}}\\ &=-\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {\left (5 \left (b^2-4 a c\right )^5 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 )}{17556 c^4 \sqrt {a+b x+c x^2}}\\ &=-\frac {5 \left (b^2-4 a c\right )^4 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{8778 c^3}-\frac {\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{2926 c^3}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{836 c^3 d}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{114 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{5/2}}{19 c d}-\frac {5 \left (b^2-4 a c\right )^{21/4} 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 )}{17556 c^4 \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 = 10.28, size = 161, normalized size = 0.50 \begin {gather*} \frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (3 (b+2 c x)^2 (a+x (b+c x))^3-2 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))^3+\frac {\left (b^2-4 a c\right )^3 \, _2F_1\left (-\frac {5}{2},\frac {1}{4};\frac {5}{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 )\right )}{57 (b+2 c x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(4*(d*(b + 2*c*x))^(7/2)*Sqrt[a + x*(b + c*x)]*(3*(b + 2*c*x)^2*(a + x*(b + c*x))^3 - 2*(a - b^2/(4*c))*c*(2*(
a + x*(b + c*x))^3 + ((b^2 - 4*a*c)^3*Hypergeometric2F1[-5/2, 1/4, 5/4, (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)]))))/(57*(b + 2*c*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1343\) vs. \(2(275)=550\).
time = 0.80, size = 1344, normalized size = 4.19 Too large to display

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/35112*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(1/2)*d^3*(59136*c^11*x^11+5120*((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^5*c^5-180*a^2*b^7*c^2-6400*((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^4*b^2*c^4+3200*((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*b^4*c^3-800*((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^6*c^2+100*((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^8*c+89168*b^6*c^5*x^5+10036*b^7*c^4*x^4-4096*a^4*c^7*x^3-4*b^8*c^3*x^3+10*b^9*c^2*x^2-10240*a^5*c^6
*x+10*b^10*c*x-5120*a^5*b*c^5+5888*a^4*b^3*c^4+1256*a^3*b^5*c^3+10*a*b^9*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^10+325248*b*c^10*x^10+216832*a*c^10*x^9+758912*b^2*c^9*x^9+975744*b^3*c^8*x^8+277760*a^2*c^9*x^7+749168*
b^4*c^7*x^7+345352*b^5*c^6*x^6+126208*a^3*c^8*x^5+975744*a*b*c^9*x^8+1812608*a*b^2*c^8*x^7+972160*a^2*b*c^8*x^
6+1790656*a*b^3*c^7*x^6+1363584*a^2*b^2*c^7*x^5+1002096*a*b^4*c^6*x^5+315520*a^3*b*c^7*x^4+978560*a^2*b^3*c^6*
x^4+305336*a*b^5*c^5*x^4+319616*a^3*b^2*c^6*x^3+369424*a^2*b^4*c^5*x^3+40208*a*b^6*c^4*x^3-6144*a^4*b*c^6*x^2+
163904*a^3*b^3*c^5*x^2+61656*a^2*b^5*c^4*x^2-192*a*b^7*c^3*x^2+9728*a^4*b^2*c^5*x+36112*a^3*b^4*c^4*x+1248*a^2
*b^6*c^3*x-180*a*b^8*c^2*x)/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 \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)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(5/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.81, size = 404, normalized size = 1.26 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\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 (14784 \, c^{10} d^{3} x^{8} + 59136 \, b c^{9} d^{3} x^{7} + 4928 \, {\left (19 \, b^{2} c^{8} + 8 \, a c^{9}\right )} d^{3} x^{6} + 14784 \, {\left (5 \, b^{3} c^{7} + 8 \, a b c^{8}\right )} d^{3} x^{5} + 28 \, {\left (1057 \, b^{4} c^{6} + 4744 \, a b^{2} c^{7} + 1072 \, a^{2} c^{8}\right )} d^{3} x^{4} + 56 \, {\left (89 \, b^{5} c^{5} + 1224 \, a b^{3} c^{6} + 1072 \, a^{2} b c^{7}\right )} d^{3} x^{3} + 6 \, {\left (3 \, b^{6} c^{4} + 2456 \, a b^{4} c^{5} + 7312 \, a^{2} b^{2} c^{6} + 256 \, a^{3} c^{7}\right )} d^{3} x^{2} - 2 \, {\left (5 \, b^{7} c^{3} - 88 \, a b^{5} c^{4} - 6928 \, a^{2} b^{3} c^{5} - 768 \, a^{3} b c^{6}\right )} d^{3} x + {\left (5 \, b^{8} c^{2} - 90 \, a b^{6} c^{3} + 628 \, a^{2} b^{4} c^{4} + 2944 \, a^{3} b^{2} c^{5} - 2560 \, a^{4} c^{6}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{35112 \, c^{5}} \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)^(5/2),x, algorithm="fricas")

[Out]

-1/35112*(5*sqrt(2)*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*
sqrt(c^2*d)*d^3*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) - 2*(14784*c^10*d^3*x^8 + 59136*b
*c^9*d^3*x^7 + 4928*(19*b^2*c^8 + 8*a*c^9)*d^3*x^6 + 14784*(5*b^3*c^7 + 8*a*b*c^8)*d^3*x^5 + 28*(1057*b^4*c^6
+ 4744*a*b^2*c^7 + 1072*a^2*c^8)*d^3*x^4 + 56*(89*b^5*c^5 + 1224*a*b^3*c^6 + 1072*a^2*b*c^7)*d^3*x^3 + 6*(3*b^
6*c^4 + 2456*a*b^4*c^5 + 7312*a^2*b^2*c^6 + 256*a^3*c^7)*d^3*x^2 - 2*(5*b^7*c^3 - 88*a*b^5*c^4 - 6928*a^2*b^3*
c^5 - 768*a^3*b*c^6)*d^3*x + (5*b^8*c^2 - 90*a*b^6*c^3 + 628*a^2*b^4*c^4 + 2944*a^3*b^2*c^5 - 2560*a^4*c^6)*d^
3)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/c^5

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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}} \left (a + b x + c x^{2}\right )^{\frac {5}{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)**(5/2),x)

[Out]

Integral((d*(b + 2*c*x))**(7/2)*(a + b*x + c*x**2)**(5/2), 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)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,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)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^(5/2), x)

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