3.337 \(\int \frac{x^m (c+d x^2)^3}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{2 a b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^3 (m+1)}-\frac{d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac{x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.224249, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {468, 570, 364} \[ -\frac{d x^{m+1} \left (a^2 d^2 (m+5)-3 a b c d (m+3)+2 b^2 c^2 (m+1)\right )}{2 a b^3 (m+1)}+\frac{x^{m+1} (b c-a d)^2 (a d (m+5)+b (c-c m)) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^3 (m+1)}-\frac{d^2 x^{m+3} (b c (m+3)-a d (m+5))}{2 a b^2 (m+3)}+\frac{x^{m+1} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{x^m \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=\frac{(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{x^m \left (c+d x^2\right ) \left (-c (b c (1-m)+a d (1+m))+d (b c (3+m)-a d (5+m)) x^2\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac{\int \left (\frac{d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^m}{b^2}+\frac{d^2 (b c (3+m)-a d (5+m)) x^{2+m}}{b}+\frac{\left (-b^3 c^3-3 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3+b^3 c^3 m-3 a b^2 c^2 d m+3 a^2 b c d^2 m-a^3 d^3 m\right ) x^m}{b^2 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac{d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac{d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac{(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}-\frac{\left (-b^3 c^3-3 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3+b^3 c^3 m-3 a b^2 c^2 d m+3 a^2 b c d^2 m-a^3 d^3 m\right ) \int \frac{x^m}{a+b x^2} \, dx}{2 a b^3}\\ &=-\frac{d \left (2 b^2 c^2 (1+m)-3 a b c d (3+m)+a^2 d^2 (5+m)\right ) x^{1+m}}{2 a b^3 (1+m)}-\frac{d^2 (b c (3+m)-a d (5+m)) x^{3+m}}{2 a b^2 (3+m)}+\frac{(b c-a d) x^{1+m} \left (c+d x^2\right )^2}{2 a b \left (a+b x^2\right )}+\frac{(b c-a d)^2 (b c (1-m)+a d (5+m)) x^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{2 a^2 b^3 (1+m)}\\ \end{align*}

Mathematica [C]  time = 4.78303, size = 2524, normalized size = 12.56 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^(1 + m)*(a*(945 + 744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(-47 + 52*m + 6*m^2 + 4*m^3 + m^4) + 3*c^2*d*(1 + m
)^4*x^2 + 3*c*d^2*(1 + m)^4*x^4 + d^3*(1 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (1 + m)/2] - 3*a*(945 +
744*m + 206*m^2 + 24*m^3 + m^4)*(c^3*(3 + m)^4 + 3*c^2*d*(65 + 92*m + 54*m^2 + 12*m^3 + m^4)*x^2 + 3*c*d^2*(3
+ m)^4*x^4 + d^3*(3 + m)^4*x^6)*HurwitzLerchPhi[-((b*x^2)/a), 1, (3 + m)/2] + 1771875*a*c^3*HurwitzLerchPhi[-(
(b*x^2)/a), 1, (5 + m)/2] + 2812500*a*c^3*m*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1927500*a*c^3*m^2*Hu
rwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*c^3*m^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 178
050*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (
5 + m)/2] + 2508*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 132*a*c^3*m^7*HurwitzLerchPhi[-((b*x^
2)/a), 1, (5 + m)/2] + 3*a*c^3*m^8*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5315625*a*c^2*d*x^2*HurwitzLe
rchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8437500*a*c^2*d*m*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 57825
00*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2236500*a*c^2*d*m^3*x^2*HurwitzLerchPhi[-((b*
x^2)/a), 1, (5 + m)/2] + 534150*a*c^2*d*m^4*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80676*a*c^2*d*m^
5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 7524*a*c^2*d*m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 +
m)/2] + 396*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 9*a*c^2*d*m^8*x^2*HurwitzLerchPhi[-
((b*x^2)/a), 1, (5 + m)/2] + 5723865*a*c*d^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 8894988*a*c*d^2
*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 5978628*a*c*d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1,
(5 + m)/2] + 2276532*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 538038*a*c*d^2*m^4*x^4*Hurw
itzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 80820*a*c*d^2*m^5*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] +
7524*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 396*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-((b*x^
2)/a), 1, (5 + m)/2] + 9*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 1771875*a*d^3*x^6*Hurwi
tzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2812500*a*d^3*m*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 192
7500*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 745500*a*d^3*m^3*x^6*HurwitzLerchPhi[-((b*x^2
)/a), 1, (5 + m)/2] + 178050*a*d^3*m^4*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 26892*a*d^3*m^5*x^6*H
urwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 2508*a*d^3*m^6*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] +
132*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (5 + m)/2] + 3*a*d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2)/a),
1, (5 + m)/2] - 2268945*a*c^3*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3082884*a*c^3*m*HurwitzLerchPhi[-(
(b*x^2)/a), 1, (7 + m)/2] - 1793204*a*c^3*m^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 585452*a*c^3*m^3*H
urwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 117670*a*c^3*m^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 14
940*a*c^3*m^5*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*c^3*m^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7
+ m)/2] - 52*a*c^3*m^7*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - a*c^3*m^8*HurwitzLerchPhi[-((b*x^2)/a),
1, (7 + m)/2] - 6806835*a*c^2*d*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c^2*d*m*x^2*Hurwit
zLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 5379612*a*c^2*d*m^2*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] -
1756356*a*c^2*d*m^3*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 353010*a*c^2*d*m^4*x^2*HurwitzLerchPhi[-
((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c^2*d*m^5*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*c^2*d*
m^6*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 156*a*c^2*d*m^7*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7
+ m)/2] - 3*a*c^2*d*m^8*x^2*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 6806835*a*c*d^2*x^4*HurwitzLerchPhi[
-((b*x^2)/a), 1, (7 + m)/2] - 9248652*a*c*d^2*m*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 5379612*a*c*
d^2*m^2*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1756356*a*c*d^2*m^3*x^4*HurwitzLerchPhi[-((b*x^2)/a)
, 1, (7 + m)/2] - 353010*a*c*d^2*m^4*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 44820*a*c*d^2*m^5*x^4*H
urwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3516*a*c*d^2*m^6*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2]
- 156*a*c*d^2*m^7*x^4*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 3*a*c*d^2*m^8*x^4*HurwitzLerchPhi[-((b*x^2
)/a), 1, (7 + m)/2] - 2042145*a*d^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 2858964*a*d^3*m*x^6*Hurw
itzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1708052*a*d^3*m^2*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] -
569804*a*d^3*m^3*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 116278*a*d^3*m^4*x^6*HurwitzLerchPhi[-((b*x
^2)/a), 1, (7 + m)/2] - 14892*a*d^3*m^5*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 1172*a*d^3*m^6*x^6*H
urwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - 52*a*d^3*m^7*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] - a*
d^3*m^8*x^6*HurwitzLerchPhi[-((b*x^2)/a), 1, (7 + m)/2] + 1536*b*c^3*x^2*HypergeometricPFQ[{2, 2, 2, 2, 3/2 +
m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c^2*d*x^4*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1
, 1, 11/2 + m/2}, -((b*x^2)/a)] + 4608*b*c*d^2*x^6*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 +
m/2}, -((b*x^2)/a)] + 1536*b*d^3*x^8*HypergeometricPFQ[{2, 2, 2, 2, 3/2 + m/2}, {1, 1, 1, 11/2 + m/2}, -((b*x
^2)/a)]))/(192*a^3*(3 + m)*(5 + m)*(7 + m)*(9 + m))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{2}+c \right ) ^{3}{x}^{m}}{ \left ( b{x}^{2}+a \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d^{3} x^{6} + 3 \, c d^{2} x^{4} + 3 \, c^{2} d x^{2} + c^{3}\right )} x^{m}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*x^m/(b^2*x^4 + 2*a*b*x^2 + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \left (c + d x^{2}\right )^{3}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Integral(x**m*(c + d*x**2)**3/(a + b*x**2)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{2} + c\right )}^{3} x^{m}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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