\(\int \frac {(a+b x^2)^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx\) [828]

Optimal result

Integrand size = 26, antiderivative size = 213 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2 \left (7 b^2 c^2+a d (14 b c-a d)\right ) \sqrt {x} \sqrt {c+d x^2}}{21 c^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {2 \left (7 b^2 c^2+a d (14 b c-a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt {c+d x^2}} \]

[Out]

-2/7*a^2*(d*x^2+c)^(3/2)/c/x^(7/2)-2/21*a*(-a*d+14*b*c)*(d*x^2+c)^(3/2)/c^2/x^(3/2)+2/21*(7*b^2*c^2+a*d*(-a*d+
14*b*c))*x^(1/2)*(d*x^2+c)^(1/2)/c^2+2/21*(7*b^2*c^2+a*d*(-a*d+14*b*c))*(cos(2*arctan(d^(1/4)*x^(1/2)/c^(1/4))
)^2)^(1/2)/cos(2*arctan(d^(1/4)*x^(1/2)/c^(1/4)))*EllipticF(sin(2*arctan(d^(1/4)*x^(1/2)/c^(1/4))),1/2*2^(1/2)
)*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(5/4)/d^(1/4)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {473, 464, 285, 335, 226} \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac {2 \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (14 b c-a d)+7 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{21 c^{5/4} \sqrt [4]{d} \sqrt {c+d x^2}}+\frac {2}{21} \sqrt {x} \sqrt {c+d x^2} \left (\frac {a d (14 b c-a d)}{c^2}+7 b^2\right )-\frac {2 a \left (c+d x^2\right )^{3/2} (14 b c-a d)}{21 c^2 x^{3/2}} \]

[In]

Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*(7*b^2 + (a*d*(14*b*c - a*d))/c^2)*Sqrt[x]*Sqrt[c + d*x^2])/21 - (2*a^2*(c + d*x^2)^(3/2))/(7*c*x^(7/2)) -
(2*a*(14*b*c - a*d)*(c + d*x^2)^(3/2))/(21*c^2*x^(3/2)) + (2*(7*b^2*c^2 + a*d*(14*b*c - a*d))*(Sqrt[c] + Sqrt[
d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(21*c^(5/
4)*d^(1/4)*Sqrt[c + d*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 285

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

Rule 464

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

Rule 473

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}+\frac {2 \int \frac {\left (\frac {1}{2} a (14 b c-a d)+\frac {7}{2} b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{5/2}} \, dx}{7 c} \\ & = -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {1}{7} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{\sqrt {x}} \, dx \\ & = \frac {2}{21} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \sqrt {x} \sqrt {c+d x^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {1}{21} \left (2 c \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right )\right ) \int \frac {1}{\sqrt {x} \sqrt {c+d x^2}} \, dx \\ & = \frac {2}{21} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \sqrt {x} \sqrt {c+d x^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {1}{21} \left (4 c \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{21} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \sqrt {x} \sqrt {c+d x^2}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{7 c x^{7/2}}-\frac {2 a (14 b c-a d) \left (c+d x^2\right )^{3/2}}{21 c^2 x^{3/2}}+\frac {2 c^{3/4} \left (7 b^2+\frac {a d (14 b c-a d)}{c^2}\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{21 \sqrt [4]{d} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2 \left (\left (c+d x^2\right ) \left (-14 a b c x^2+7 b^2 c x^4-a^2 \left (3 c+2 d x^2\right )\right )+\frac {2 i \left (7 b^2 c^2+14 a b c d-a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^{9/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}\right )}{21 c x^{7/2} \sqrt {c+d x^2}} \]

[In]

Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(9/2),x]

[Out]

(2*((c + d*x^2)*(-14*a*b*c*x^2 + 7*b^2*c*x^4 - a^2*(3*c + 2*d*x^2)) + ((2*I)*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2
)*Sqrt[1 + c/(d*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/Sqrt[(I*Sqrt[c])/Sq
rt[d]]))/(21*c*x^(7/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-7 b^{2} c \,x^{4}+2 a^{2} d \,x^{2}+14 a b c \,x^{2}+3 a^{2} c \right )}{21 x^{\frac {7}{2}} c}-\frac {2 \left (a^{2} d^{2}-14 a b c d -7 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x \left (d \,x^{2}+c \right )}}{21 c d \sqrt {d \,x^{3}+c x}\, \sqrt {x}\, \sqrt {d \,x^{2}+c}}\)	\(208\)
elliptic	\(\frac {\sqrt {x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d \,x^{3}+c x}}{7 x^{4}}-\frac {4 a \left (a d +7 b c \right ) \sqrt {d \,x^{3}+c x}}{21 c \,x^{2}}+\frac {2 b^{2} \sqrt {d \,x^{3}+c x}}{3}+\frac {\left (2 a b d +\frac {2 b^{2} c}{3}-\frac {2 d a \left (a d +7 b c \right )}{21 c}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d \,x^{3}+c x}}\right )}{\sqrt {x}\, \sqrt {d \,x^{2}+c}}\)	\(223\)
default	\(-\frac {2 \left (\sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} x^{3}-14 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d \,x^{3}-7 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2} x^{3}-7 b^{2} c \,d^{2} x^{6}+2 a^{2} d^{3} x^{4}+14 a b c \,d^{2} x^{4}-7 b^{2} c^{2} d \,x^{4}+5 a^{2} c \,d^{2} x^{2}+14 a b \,c^{2} d \,x^{2}+3 a^{2} c^{2} d \right )}{21 \sqrt {d \,x^{2}+c}\, x^{\frac {7}{2}} d c}\)	\(385\)

[In]

int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/21*(d*x^2+c)^(1/2)*(-7*b^2*c*x^4+2*a^2*d*x^2+14*a*b*c*x^2+3*a^2*c)/x^(7/2)/c-2/21*(a^2*d^2-14*a*b*c*d-7*b^2
*c^2)/c*(-c*d)^(1/2)/d*((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x-(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*
(-x/(-c*d)^(1/2)*d)^(1/2)/(d*x^3+c*x)^(1/2)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2))*(
x*(d*x^2+c))^(1/2)/x^(1/2)/(d*x^2+c)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {2 \, {\left (2 \, {\left (7 \, b^{2} c^{2} + 14 \, a b c d - a^{2} d^{2}\right )} \sqrt {d} x^{4} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (7 \, b^{2} c d x^{4} - 3 \, a^{2} c d - 2 \, {\left (7 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )}}{21 \, c d x^{4}} \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="fricas")

[Out]

2/21*(2*(7*b^2*c^2 + 14*a*b*c*d - a^2*d^2)*sqrt(d)*x^4*weierstrassPInverse(-4*c/d, 0, x) + (7*b^2*c*d*x^4 - 3*
a^2*c*d - 2*(7*a*b*c*d + a^2*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(x))/(c*d*x^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\frac {a^{2} \sqrt {c} \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac {7}{2}} \Gamma \left (- \frac {3}{4}\right )} + \frac {a b \sqrt {c} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {b^{2} \sqrt {c} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(9/2),x)

[Out]

a**2*sqrt(c)*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4,), d*x**2*exp_polar(I*pi)/c)/(2*x**(7/2)*gamma(-3/4)) + a*b
*sqrt(c)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), d*x**2*exp_polar(I*pi)/c)/(x**(3/2)*gamma(1/4)) + b**2*sqrt(c
)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), d*x**2*exp_polar(I*pi)/c)/(2*gamma(5/4))

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {9}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {9}{2}}} \,d x } \]

[In]

integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(9/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{9/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^{9/2}} \,d x \]

[In]

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^(9/2),x)

[Out]

int(((a + b*x^2)^2*(c + d*x^2)^(1/2))/x^(9/2), x)