3.14.90 \(\int \frac {(b d+2 c d x)^{11/2}}{(a+b x+c x^2)^{5/2}} \, dx\) [1390]

Optimal. Leaf size=196 \[ -\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\sqrt {a+b x+c x^2}} \]

[Out]

-2/3*d*(2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^(3/2)-12*c*d^3*(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(1/2)+80*c^2*d^5*(2*
c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)+40*c*(-4*a*c+b^2)^(5/4)*d^(11/2)*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*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 196, 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 = {700, 706, 705, 703, 227} \begin {gather*} \frac {40 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \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 )}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d*(b*d + 2*c*d*x)^(9/2))/(3*(a + b*x + c*x^2)^(3/2)) - (12*c*d^3*(b*d + 2*c*d*x)^(5/2))/Sqrt[a + b*x + c*x
^2] + 80*c^2*d^5*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2] + (40*c*(b^2 - 4*a*c)^(5/4)*d^(11/2)*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])/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 700

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*(d + e*x)^(m - 1)*(
(a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Dist[d*e*((m - 1)/(b*(p + 1))), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2
*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && 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 \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (6 c d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\left (60 c^2 d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\left (20 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (20 c^2 \left (b^2-4 a c\right ) d^6 \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}{\sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (40 c \left (b^2-4 a c\right ) d^5 \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 )}{\sqrt {a+b x+c x^2}}\\ &=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \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 )}{\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 = 10.15, size = 201, normalized size = 1.03 \begin {gather*} \frac {2 d^5 \sqrt {d (b+2 c x)} \left (-b^4-26 b^3 c x+6 b^2 c \left (-3 a+c x^2\right )+8 b c^2 x \left (21 a+8 c x^2\right )+8 c^2 \left (15 a^2+21 a c x^2+4 c^2 x^4\right )-60 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \left (4 a^2 c-b^2 x (b+c x)+a \left (-b^2+4 b c x+4 c^2 x^2\right )\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 (a+x (b+c x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*d^5*Sqrt[d*(b + 2*c*x)]*(-b^4 - 26*b^3*c*x + 6*b^2*c*(-3*a + c*x^2) + 8*b*c^2*x*(21*a + 8*c*x^2) + 8*c^2*(1
5*a^2 + 21*a*c*x^2 + 4*c^2*x^4) - 60*c*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*(4*a^2*c - b^2*x*(b + c*x) +
a*(-b^2 + 4*b*c*x + 4*c^2*x^2))*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(3*(a + x*(b
+ c*x))^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(168)=336\).
time = 0.93, size = 958, normalized size = 4.89 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(120*(-4*a*c+b^2)^(1/2)*((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))*a*c^3*x^2-30*(-4*a*c+b^2)^(1/2)*((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))*b^2*c^2*x^2+1
20*(-4*a*c+b^2)^(1/2)*((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))*a*b*c^2*x-30*(-4*a*c+b^2)^(1/2)*((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))*b^3*c*x-64*c^5*x^5+1
20*(-4*a*c+b^2)^(1/2)*((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))*a^2*c^2-30*(-4*a*c+b^2)^(1/2)*((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))*a*b^2*c-160*b*c^4*x^4-
336*a*c^4*x^3-76*b^2*c^3*x^3-504*a*b*c^3*x^2+46*b^3*c^2*x^2-240*a^2*c^3*x-132*b^2*c^2*a*x+28*c*b^4*x-120*a^2*b
*c^2+18*a*b^3*c+b^5)*d^5*(d*(2*c*x+b))^(1/2)/(2*c*x+b)/(c*x^2+b*x+a)^(3/2)

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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)^(11/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

integrate((2*c*d*x + b*d)^(11/2)/(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 = 0.94, size = 297, normalized size = 1.52 \begin {gather*} \frac {2 \, {\left (30 \, \sqrt {2} {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{4} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{5} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{5} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (32 \, c^{4} d^{5} x^{4} + 64 \, b c^{3} d^{5} x^{3} + 6 \, {\left (b^{2} c^{2} + 28 \, a c^{3}\right )} d^{5} x^{2} - 2 \, {\left (13 \, b^{3} c - 84 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 18 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/3*(30*sqrt(2)*((b^2*c^2 - 4*a*c^3)*d^5*x^4 + 2*(b^3*c - 4*a*b*c^2)*d^5*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d
^5*x^2 + 2*(a*b^3 - 4*a^2*b*c)*d^5*x + (a^2*b^2 - 4*a^3*c)*d^5)*sqrt(c^2*d)*weierstrassPInverse((b^2 - 4*a*c)/
c^2, 0, 1/2*(2*c*x + b)/c) + (32*c^4*d^5*x^4 + 64*b*c^3*d^5*x^3 + 6*(b^2*c^2 + 28*a*c^3)*d^5*x^2 - 2*(13*b^3*c
- 84*a*b*c^2)*d^5*x - (b^4 + 18*a*b^2*c - 120*a^2*c^2)*d^5)*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a))/(c^2*x
^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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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)^(11/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{11/2}}{{\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)^(11/2)/(a + b*x + c*x^2)^(5/2),x)

[Out]

int((b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^(5/2), x)

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