3.7.36 \(\int \frac {1}{(c x)^{7/2} (a+b x^2)^{5/2}} \, dx\) [636]

Optimal. Leaf size=362 \[ \frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {77 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{10 a^4 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{10 a^{15/4} c^{7/2} \sqrt {a+b x^2}}-\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{20 a^{15/4} c^{7/2} \sqrt {a+b x^2}} \]

[Out]

1/3/a/c/(c*x)^(5/2)/(b*x^2+a)^(3/2)+11/6/a^2/c/(c*x)^(5/2)/(b*x^2+a)^(1/2)-77/30*(b*x^2+a)^(1/2)/a^3/c/(c*x)^(
5/2)+77/10*b*(b*x^2+a)^(1/2)/a^4/c^3/(c*x)^(1/2)-77/10*b^(3/2)*(c*x)^(1/2)*(b*x^2+a)^(1/2)/a^4/c^4/(a^(1/2)+x*
b^(1/2))+77/10*b^(5/4)*(cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)
^(1/2)/a^(1/4)/c^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*
b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(15/4)/c^(7/2)/(b*x^2+a)^(1/2)-77/20*b^(5/4)*(cos(2*arctan(
b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2)))*EllipticF(si
n(2*arctan(b^(1/4)*(c*x)^(1/2)/a^(1/4)/c^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2
))^2)^(1/2)/a^(15/4)/c^(7/2)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {296, 331, 335, 311, 226, 1210} \begin {gather*} -\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{20 a^{15/4} c^{7/2} \sqrt {a+b x^2}}+\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{10 a^{15/4} c^{7/2} \sqrt {a+b x^2}}-\frac {77 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{10 a^4 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}+\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x]

[Out]

1/(3*a*c*(c*x)^(5/2)*(a + b*x^2)^(3/2)) + 11/(6*a^2*c*(c*x)^(5/2)*Sqrt[a + b*x^2]) - (77*Sqrt[a + b*x^2])/(30*
a^3*c*(c*x)^(5/2)) + (77*b*Sqrt[a + b*x^2])/(10*a^4*c^3*Sqrt[c*x]) - (77*b^(3/2)*Sqrt[c*x]*Sqrt[a + b*x^2])/(1
0*a^4*c^4*(Sqrt[a] + Sqrt[b]*x)) + (77*b^(5/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]
*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(10*a^(15/4)*c^(7/2)*Sqrt[a + b*x^2]) - (77*
b^(5/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c*x])
/(a^(1/4)*Sqrt[c])], 1/2])/(20*a^(15/4)*c^(7/2)*Sqrt[a + b*x^2])

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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11 \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx}{6 a}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}+\frac {77 \int \frac {1}{(c x)^{7/2} \sqrt {a+b x^2}} \, dx}{12 a^2}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}-\frac {(77 b) \int \frac {1}{(c x)^{3/2} \sqrt {a+b x^2}} \, dx}{20 a^3 c^2}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {\left (77 b^2\right ) \int \frac {\sqrt {c x}}{\sqrt {a+b x^2}} \, dx}{20 a^4 c^4}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {\left (77 b^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{10 a^4 c^5}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {\left (77 b^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{10 a^{7/2} c^4}+\frac {\left (77 b^{3/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} c}}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{10 a^{7/2} c^4}\\ &=\frac {1}{3 a c (c x)^{5/2} \left (a+b x^2\right )^{3/2}}+\frac {11}{6 a^2 c (c x)^{5/2} \sqrt {a+b x^2}}-\frac {77 \sqrt {a+b x^2}}{30 a^3 c (c x)^{5/2}}+\frac {77 b \sqrt {a+b x^2}}{10 a^4 c^3 \sqrt {c x}}-\frac {77 b^{3/2} \sqrt {c x} \sqrt {a+b x^2}}{10 a^4 c^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{10 a^{15/4} c^{7/2} \sqrt {a+b x^2}}-\frac {77 b^{5/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{20 a^{15/4} c^{7/2} \sqrt {a+b 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 = 10.03, size = 59, normalized size = 0.16 \begin {gather*} -\frac {2 x \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {5}{4},\frac {5}{2};-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 a^2 (c x)^{7/2} \sqrt {a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x]

[Out]

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

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Maple [A]
time = 0.10, size = 410, normalized size = 1.13

method	result	size
elliptic	\(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {x \sqrt {b c \,x^{3}+a c x}}{3 a^{3} c^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {5 b^{2} x^{2}}{2 c^{3} a^{4} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {2 \sqrt {b c \,x^{3}+a c x}}{5 a^{3} c^{4} x^{3}}+\frac {26 \left (c \,x^{2} b +a c \right ) b}{5 a^{4} c^{4} \sqrt {x \left (c \,x^{2} b +a c \right )}}-\frac {77 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{20 a^{4} c^{3} \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(312\)
default	\(-\frac {462 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}-231 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{4}+462 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-231 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}-462 b^{3} x^{6}-770 a \,b^{2} x^{4}-264 a^{2} b \,x^{2}+24 a^{3}}{60 x^{2} a^{4} c^{3} \sqrt {c x}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(410\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-13 b \,x^{2}+a \right )}{5 a^{4} x^{2} c^{3} \sqrt {c x}}-\frac {b^{2} \left (\frac {13 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b c \,x^{3}+a c x}}-10 a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b c \,x^{3}+a c x}}\right )-5 a^{2} \left (\frac {x \sqrt {b c \,x^{3}+a c x}}{3 a c \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {x^{2}}{2 a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{2} b \sqrt {b c \,x^{3}+a c x}}\right )\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{5 a^{4} c^{3} \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\)	\(650\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/60*(462*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*
b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*x^4-231*((b*x+(-a*b)^(1/2
))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((
b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2*x^4+462*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/
2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/
2))^(1/2),1/2*2^(1/2))*a^2*b*x^2-231*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*
b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b
*x^2-462*b^3*x^6-770*a*b^2*x^4-264*a^2*b*x^2+24*a^3)/x^2/a^4/c^3/(c*x)^(1/2)/(b*x^2+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(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(7/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.25, size = 137, normalized size = 0.38 \begin {gather*} \frac {231 \, {\left (b^{3} x^{7} + 2 \, a b^{2} x^{5} + a^{2} b x^{3}\right )} \sqrt {b c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (231 \, b^{3} x^{6} + 385 \, a b^{2} x^{4} + 132 \, a^{2} b x^{2} - 12 \, a^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{30 \, {\left (a^{4} b^{2} c^{4} x^{7} + 2 \, a^{5} b c^{4} x^{5} + a^{6} c^{4} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/30*(231*(b^3*x^7 + 2*a*b^2*x^5 + a^2*b*x^3)*sqrt(b*c)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b,
0, x)) + (231*b^3*x^6 + 385*a*b^2*x^4 + 132*a^2*b*x^2 - 12*a^3)*sqrt(b*x^2 + a)*sqrt(c*x))/(a^4*b^2*c^4*x^7 +
2*a^5*b*c^4*x^5 + a^6*c^4*x^3)

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Sympy [C] Result contains complex when optimal does not.
time = 27.71, size = 51, normalized size = 0.14 \begin {gather*} \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} c^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(7/2)/(b*x**2+a)**(5/2),x)

[Out]

gamma(-5/4)*hyper((-5/4, 5/2), (-1/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*c**(7/2)*x**(5/2)*gamma(-1/4))

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

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(c*x)^(7/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(7/2)*(a + b*x^2)^(5/2)),x)

[Out]

int(1/((c*x)^(7/2)*(a + b*x^2)^(5/2)), x)

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