3.7.29 \(\int \frac {(c x)^{7/2}}{(a+b x^2)^{5/2}} \, dx\) [629]

Optimal. Leaf size=155 \[ -\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {5 c^{7/2} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right )|\frac {1}{2}\right )}{12 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}} \]

[Out]

-1/3*c*(c*x)^(5/2)/b/(b*x^2+a)^(3/2)-5/6*c^3*(c*x)^(1/2)/b^2/(b*x^2+a)^(1/2)+5/12*c^(7/2)*(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(sin(2*ar
ctan(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^(1/4)/b^(9/4)/(b*x^2+a)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {294, 335, 226} \begin {gather*} \frac {5 c^{7/2} \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 )}{12 \sqrt [4]{a} b^{9/4} \sqrt {a+b x^2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x)^(7/2)/(a + b*x^2)^(5/2),x]

[Out]

-1/3*(c*(c*x)^(5/2))/(b*(a + b*x^2)^(3/2)) - (5*c^3*Sqrt[c*x])/(6*b^2*Sqrt[a + b*x^2]) + (5*c^(7/2)*(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])/(12*a^(1/4)*b^(9/4)*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 294

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

Rubi steps

\begin {align*} \int \frac {(c x)^{7/2}}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {(c x)^{3/2}}{\left (a+b x^2\right )^{3/2}} \, dx}{6 b}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {\left (5 c^4\right ) \int \frac {1}{\sqrt {c x} \sqrt {a+b x^2}} \, dx}{12 b^2}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{6 b^2}\\ &=-\frac {c (c x)^{5/2}}{3 b \left (a+b x^2\right )^{3/2}}-\frac {5 c^3 \sqrt {c x}}{6 b^2 \sqrt {a+b x^2}}+\frac {5 c^{7/2} \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 )}{12 \sqrt [4]{a} b^{9/4} \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 = 10.05, size = 80, normalized size = 0.52 \begin {gather*} \frac {c^3 \sqrt {c x} \left (-5 a-7 b x^2+5 \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^2}{a}\right )\right )}{6 b^2 \left (a+b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*Sqrt[c*x]*(-5*a - 7*b*x^2 + 5*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^2)/
a)]))/(6*b^2*(a + b*x^2)^(3/2))

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Maple [A]
time = 0.08, size = 219, normalized size = 1.41

method	result	size
elliptic	\(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {a \,c^{3} \sqrt {b c \,x^{3}+a c x}}{3 b^{4} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {7 c^{4} x}{6 b^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b c x}}+\frac {5 c^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{12 b^{3} \sqrt {b c \,x^{3}+a c x}}\right )}{c x \sqrt {b \,x^{2}+a}}\)	\(205\)
default	\(\frac {\left (5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, b \,x^{2}+5 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a -14 b^{2} x^{3}-10 a b x \right ) c^{3} \sqrt {c x}}{12 x \,b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(219\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(5*((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)^(1/2)*b*x^2+5*((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)*Elliptic
F(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a-14*b^2*x^3-10*a*b*x)*c^3/x*(c*x)^(1/2)/b
^3/(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((c*x)^(7/2)/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 108, normalized size = 0.70 \begin {gather*} \frac {5 \, {\left (b^{2} c^{3} x^{4} + 2 \, a b c^{3} x^{2} + a^{2} c^{3}\right )} \sqrt {b c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) - {\left (7 \, b^{2} c^{3} x^{2} + 5 \, a b c^{3}\right )} \sqrt {b x^{2} + a} \sqrt {c x}}{6 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(5*(b^2*c^3*x^4 + 2*a*b*c^3*x^2 + a^2*c^3)*sqrt(b*c)*weierstrassPInverse(-4*a/b, 0, x) - (7*b^2*c^3*x^2 +
5*a*b*c^3)*sqrt(b*x^2 + a)*sqrt(c*x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)

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Sympy [C] Result contains complex when optimal does not.
time = 13.84, size = 44, normalized size = 0.28 \begin {gather*} \frac {c^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {5}{2} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**(7/2)*x**(9/2)*gamma(9/4)*hyper((9/4, 5/2), (13/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(13/4))

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\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((c*x)^(7/2)/(a + b*x^2)^(5/2),x)

[Out]

int((c*x)^(7/2)/(a + b*x^2)^(5/2), x)

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