3.3.44 \(\int \frac {x^{13/2} (A+B x^2)}{\sqrt {b x^2+c x^4}} \, dx\) [244]

Optimal. Leaf size=243 \[ -\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {b^{11/4} (13 b B-15 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{17/4} \sqrt {b x^2+c x^4}} \]

[Out]

6/385*b*(-15*A*c+13*B*b)*x^(3/2)*(c*x^4+b*x^2)^(1/2)/c^3-2/165*(-15*A*c+13*B*b)*x^(7/2)*(c*x^4+b*x^2)^(1/2)/c^
2+2/15*B*x^(11/2)*(c*x^4+b*x^2)^(1/2)/c-2/77*b^2*(-15*A*c+13*B*b)*(c*x^4+b*x^2)^(1/2)/c^4/x^(1/2)+1/77*b^(11/4
)*(-15*A*c+13*B*b)*x*(cos(2*arctan(c^(1/4)*x^(1/2)/b^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/b^(1/4)))*E
llipticF(sin(2*arctan(c^(1/4)*x^(1/2)/b^(1/4))),1/2*2^(1/2))*(b^(1/2)+x*c^(1/2))*((c*x^2+b)/(b^(1/2)+x*c^(1/2)
)^2)^(1/2)/c^(17/4)/(c*x^4+b*x^2)^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2064, 2049, 2057, 335, 226} \begin {gather*} \frac {b^{11/4} 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-15 A c) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 c^{17/4} \sqrt {b x^2+c x^4}}-\frac {2 b^2 \sqrt {b x^2+c x^4} (13 b B-15 A c)}{77 c^4 \sqrt {x}}+\frac {6 b x^{3/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{385 c^3}-\frac {2 x^{7/2} \sqrt {b x^2+c x^4} (13 b B-15 A c)}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^(13/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(-2*b^2*(13*b*B - 15*A*c)*Sqrt[b*x^2 + c*x^4])/(77*c^4*Sqrt[x]) + (6*b*(13*b*B - 15*A*c)*x^(3/2)*Sqrt[b*x^2 +
c*x^4])/(385*c^3) - (2*(13*b*B - 15*A*c)*x^(7/2)*Sqrt[b*x^2 + c*x^4])/(165*c^2) + (2*B*x^(11/2)*Sqrt[b*x^2 + c
*x^4])/(15*c) + (b^(11/4)*(13*b*B - 15*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])/(77*c^(17/4)*Sqrt[b*x^2 + c*x^4])

Rule 226

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)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 335

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 2049

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 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[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 2064

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])

Rubi steps

\begin {align*} \int \frac {x^{13/2} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}-\frac {\left (2 \left (\frac {13 b B}{2}-\frac {15 A c}{2}\right )\right ) \int \frac {x^{13/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 c}\\ &=-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {(3 b (13 b B-15 A c)) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx}{55 c^2}\\ &=\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}-\frac {\left (3 b^2 (13 b B-15 A c)\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^3}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (b^3 (13 b B-15 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^4}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (b^3 (13 b B-15 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{77 c^4 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {\left (2 b^3 (13 b B-15 A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{77 c^4 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 b^2 (13 b B-15 A c) \sqrt {b x^2+c x^4}}{77 c^4 \sqrt {x}}+\frac {6 b (13 b B-15 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^3}-\frac {2 (13 b B-15 A c) x^{7/2} \sqrt {b x^2+c x^4}}{165 c^2}+\frac {2 B x^{11/2} \sqrt {b x^2+c x^4}}{15 c}+\frac {b^{11/4} (13 b B-15 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 )}{77 c^{17/4} \sqrt {b x^2+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.13, size = 143, normalized size = 0.59 \begin {gather*} \frac {2 x^{3/2} \left (-\left (\left (b+c x^2\right ) \left (195 b^3 B-7 c^3 x^4 \left (15 A+11 B x^2\right )-9 b^2 c \left (25 A+13 B x^2\right )+b c^2 x^2 \left (135 A+91 B x^2\right )\right )\right )+15 b^3 (13 b B-15 A c) \sqrt {1+\frac {c x^2}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{1155 c^4 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(13/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]

[Out]

(2*x^(3/2)*(-((b + c*x^2)*(195*b^3*B - 7*c^3*x^4*(15*A + 11*B*x^2) - 9*b^2*c*(25*A + 13*B*x^2) + b*c^2*x^2*(13
5*A + 91*B*x^2))) + 15*b^3*(13*b*B - 15*A*c)*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/b)
]))/(1155*c^4*Sqrt[x^2*(b + c*x^2)])

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Maple [A]
time = 0.40, size = 298, normalized size = 1.23

method	result	size
risch	\(\frac {2 \left (77 B \,c^{3} x^{6}+105 A \,c^{3} x^{4}-91 B b \,c^{2} x^{4}-135 A b \,c^{2} x^{2}+117 B \,b^{2} c \,x^{2}+225 A \,b^{2} c -195 B \,b^{3}\right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{1155 c^{4} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {b^{3} \left (15 A c -13 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{77 c^{5} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(239\)
default	\(-\frac {\sqrt {x}\, \left (-154 B \,c^{5} x^{9}-210 A \,c^{5} x^{7}+28 B b \,c^{4} x^{7}+225 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b^{3} c +60 A b \,c^{4} x^{5}-195 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b^{4}-52 B \,b^{2} c^{3} x^{5}-180 A \,b^{2} c^{3} x^{3}+156 B \,b^{3} c^{2} x^{3}-450 A \,b^{3} c^{2} x +390 B \,b^{4} c x \right )}{1155 \sqrt {x^{4} c +b \,x^{2}}\, c^{5}}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(13/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/1155/(c*x^4+b*x^2)^(1/2)*x^(1/2)*(-154*B*c^5*x^9-210*A*c^5*x^7+28*B*b*c^4*x^7+225*A*((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*c)^(1/2)*b^3*c+60*A*b*c^4*x^5-195*B*((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*c)^(1/2)*b^4-52*B*b^2*c^3*x^5-180*A*b^2*c^3*x^3+156*B*b^3*c^2*x
^3-450*A*b^3*c^2*x+390*B*b^4*c*x)/c^5

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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(x^(13/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^(13/2)/sqrt(c*x^4 + b*x^2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 122, normalized size = 0.50 \begin {gather*} \frac {2 \, {\left (15 \, {\left (13 \, B b^{4} - 15 \, A b^{3} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (77 \, B c^{4} x^{6} - 195 \, B b^{3} c + 225 \, A b^{2} c^{2} - 7 \, {\left (13 \, B b c^{3} - 15 \, A c^{4}\right )} x^{4} + 9 \, {\left (13 \, B b^{2} c^{2} - 15 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, c^{5} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/1155*(15*(13*B*b^4 - 15*A*b^3*c)*sqrt(c)*x*weierstrassPInverse(-4*b/c, 0, x) + (77*B*c^4*x^6 - 195*B*b^3*c +
225*A*b^2*c^2 - 7*(13*B*b*c^3 - 15*A*c^4)*x^4 + 9*(13*B*b^2*c^2 - 15*A*b*c^3)*x^2)*sqrt(c*x^4 + b*x^2)*sqrt(x
))/(c^5*x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(13/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5457 deep

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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(x^(13/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^(13/2)/sqrt(c*x^4 + b*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^{13/2}\,\left (B\,x^2+A\right )}{\sqrt {c\,x^4+b\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(1/2),x)

[Out]

int((x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(1/2), x)

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