3.232 \(\int x^{5/2} (A+B x^2) (b x^2+c x^4)^{3/2} \, dx\)

Optimal. Leaf size=321 \[ \frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]

[Out]

(-24*b^4*(13*b*B - 23*A*c)*Sqrt[b*x^2 + c*x^4])/(33649*c^4*Sqrt[x]) + (72*b^3*(13*b*B - 23*A*c)*x^(3/2)*Sqrt[b
*x^2 + c*x^4])/(168245*c^3) - (8*b^2*(13*b*B - 23*A*c)*x^(7/2)*Sqrt[b*x^2 + c*x^4])/(24035*c^2) - (4*b*(13*b*B
- 23*A*c)*x^(11/2)*Sqrt[b*x^2 + c*x^4])/(2185*c) - (2*(13*b*B - 23*A*c)*x^(7/2)*(b*x^2 + c*x^4)^(3/2))/(437*c
) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(5/2))/(23*c) + (12*b^(19/4)*(13*b*B - 23*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(
b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(33649*c^(17/4)*Sqrt[
b*x^2 + c*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.489965, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2039, 2021, 2024, 2032, 329, 220} \[ -\frac{8 b^2 x^{7/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{24035 c^2}+\frac{72 b^3 x^{3/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{168245 c^3}-\frac{24 b^4 \sqrt{b x^2+c x^4} (13 b B-23 A c)}{33649 c^4 \sqrt{x}}+\frac{12 b^{19/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (13 b B-23 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}-\frac{4 b x^{11/2} \sqrt{b x^2+c x^4} (13 b B-23 A c)}{2185 c}-\frac{2 x^{7/2} \left (b x^2+c x^4\right )^{3/2} (13 b B-23 A c)}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(-24*b^4*(13*b*B - 23*A*c)*Sqrt[b*x^2 + c*x^4])/(33649*c^4*Sqrt[x]) + (72*b^3*(13*b*B - 23*A*c)*x^(3/2)*Sqrt[b
*x^2 + c*x^4])/(168245*c^3) - (8*b^2*(13*b*B - 23*A*c)*x^(7/2)*Sqrt[b*x^2 + c*x^4])/(24035*c^2) - (4*b*(13*b*B
- 23*A*c)*x^(11/2)*Sqrt[b*x^2 + c*x^4])/(2185*c) - (2*(13*b*B - 23*A*c)*x^(7/2)*(b*x^2 + c*x^4)^(3/2))/(437*c
) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(5/2))/(23*c) + (12*b^(19/4)*(13*b*B - 23*A*c)*x*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(
b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(33649*c^(17/4)*Sqrt[
b*x^2 + c*x^4])

Rule 2039

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(d*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(b*(m + n + p*(j + n) + 1)), x] - Dist[(a*d*(m
+ j*p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1)), Int[(e*x)^m*(a*x^j + b*x^(j + n))^p, x
], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[
m + n + p*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])

Rule 2021

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b
*x^n)^p)/(c*(m + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*p + 1)), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2024

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

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int x^{5/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (2 \left (\frac{13 b B}{2}-\frac{23 A c}{2}\right )\right ) \int x^{5/2} \left (b x^2+c x^4\right )^{3/2} \, dx}{23 c}\\ &=-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{(6 b (13 b B-23 A c)) \int x^{9/2} \sqrt{b x^2+c x^4} \, dx}{437 c}\\ &=-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (4 b^2 (13 b B-23 A c)\right ) \int \frac{x^{13/2}}{\sqrt{b x^2+c x^4}} \, dx}{2185 c}\\ &=-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (36 b^3 (13 b B-23 A c)\right ) \int \frac{x^{9/2}}{\sqrt{b x^2+c x^4}} \, dx}{24035 c^2}\\ &=\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}-\frac{\left (36 b^4 (13 b B-23 A c)\right ) \int \frac{x^{5/2}}{\sqrt{b x^2+c x^4}} \, dx}{33649 c^3}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (12 b^5 (13 b B-23 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{33649 c^4}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (12 b^5 (13 b B-23 A c) x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{33649 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{\left (24 b^5 (13 b B-23 A c) x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{33649 c^4 \sqrt{b x^2+c x^4}}\\ &=-\frac{24 b^4 (13 b B-23 A c) \sqrt{b x^2+c x^4}}{33649 c^4 \sqrt{x}}+\frac{72 b^3 (13 b B-23 A c) x^{3/2} \sqrt{b x^2+c x^4}}{168245 c^3}-\frac{8 b^2 (13 b B-23 A c) x^{7/2} \sqrt{b x^2+c x^4}}{24035 c^2}-\frac{4 b (13 b B-23 A c) x^{11/2} \sqrt{b x^2+c x^4}}{2185 c}-\frac{2 (13 b B-23 A c) x^{7/2} \left (b x^2+c x^4\right )^{3/2}}{437 c}+\frac{2 B x^{3/2} \left (b x^2+c x^4\right )^{5/2}}{23 c}+\frac{12 b^{19/4} (13 b B-23 A c) x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{33649 c^{17/4} \sqrt{b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 0.207018, size = 160, normalized size = 0.5 \[ \frac{2 \sqrt{x^2 \left (b+c x^2\right )} \left (15 b^4 (13 b B-23 A c) \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{c x^2}{b}\right )-\left (b+c x^2\right )^2 \sqrt{\frac{c x^2}{b}+1} \left (-3 b^2 c \left (115 A+143 B x^2\right )+11 b c^2 x^2 \left (69 A+65 B x^2\right )-55 c^3 x^4 \left (23 A+19 B x^2\right )+195 b^3 B\right )\right )}{24035 c^4 \sqrt{x} \sqrt{\frac{c x^2}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]

[Out]

(2*Sqrt[x^2*(b + c*x^2)]*(-((b + c*x^2)^2*Sqrt[1 + (c*x^2)/b]*(195*b^3*B - 55*c^3*x^4*(23*A + 19*B*x^2) + 11*b
*c^2*x^2*(69*A + 65*B*x^2) - 3*b^2*c*(115*A + 143*B*x^2))) + 15*b^4*(13*b*B - 23*A*c)*Hypergeometric2F1[-3/2,
1/4, 5/4, -((c*x^2)/b)]))/(24035*c^4*Sqrt[x]*Sqrt[1 + (c*x^2)/b])

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 355, normalized size = 1.1 \begin{align*} -{\frac{2}{168245\, \left ( c{x}^{2}+b \right ) ^{2}{c}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -7315\,B{x}^{13}{c}^{7}-8855\,A{x}^{11}{c}^{7}-16940\,B{x}^{11}b{c}^{6}-21252\,A{x}^{9}b{c}^{6}-9933\,B{x}^{9}{b}^{2}{c}^{5}-13041\,A{x}^{7}{b}^{2}{c}^{5}+56\,B{x}^{7}{b}^{3}{c}^{4}+690\,A\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{5}c+184\,A{x}^{5}{b}^{3}{c}^{4}-390\,B\sqrt{-bc}\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){b}^{6}-104\,B{x}^{5}{b}^{4}{c}^{3}-552\,A{x}^{3}{b}^{4}{c}^{3}+312\,B{x}^{3}{b}^{5}{c}^{2}-1380\,Ax{b}^{5}{c}^{2}+780\,Bx{b}^{6}c \right ){x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)

[Out]

-2/168245*(c*x^4+b*x^2)^(3/2)/x^(7/2)/(c*x^2+b)^2*(-7315*B*x^13*c^7-8855*A*x^11*c^7-16940*B*x^11*b*c^6-21252*A
*x^9*b*c^6-9933*B*x^9*b^2*c^5-13041*A*x^7*b^2*c^5+56*B*x^7*b^3*c^4+690*A*(-b*c)^(1/2)*((c*x+(-b*c)^(1/2))/(-b*
c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b
*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b^5*c+184*A*x^5*b^3*c^4-390*B*(-b*c)^(1/2)*((c*x+(-b*c)^(1/2))/(-b
*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-
b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b^6-104*B*x^5*b^4*c^3-552*A*x^3*b^4*c^3+312*B*x^3*b^5*c^2-1380*A*
x*b^5*c^2+780*B*x*b^6*c)/c^5

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B c x^{8} +{\left (B b + A c\right )} x^{6} + A b x^{4}\right )} \sqrt{c x^{4} + b x^{2}} \sqrt{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

integral((B*c*x^8 + (B*b + A*c)*x^6 + A*b*x^4)*sqrt(c*x^4 + b*x^2)*sqrt(x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}{\left (B x^{2} + A\right )} x^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2)^(3/2)*(B*x^2 + A)*x^(5/2), x)