3.3.6 \(\int \frac {1}{(a+b x^2)^{7/2} \sqrt {c+d x^2}} \, dx\) [206]

Optimal. Leaf size=334 \[ \frac {b x \sqrt {c+d x^2}}{5 a (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 b (b c-2 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {\sqrt {b} \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} (b c-a d)^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

-1/15*(15*a^2*d^2-11*a*b*c*d+4*b^2*c^2)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1
+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*d^(1/2)*(b*x^2+a)^(1/2)/a^3/(-a*d+b*c)^3/(c*(b*x^2+a)/a/(d*x^2+c))^
(1/2)/(d*x^2+c)^(1/2)+1/5*b*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(5/2)+4/15*b*(-2*a*d+b*c)*x*(d*x^2+c)^(1/
2)/a^2/(-a*d+b*c)^2/(b*x^2+a)^(3/2)+1/15*(23*a^2*d^2-23*a*b*c*d+8*b^2*c^2)*(1/(1+b*x^2/a))^(1/2)*(1+b*x^2/a)^(
1/2)*EllipticE(x*b^(1/2)/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))*b^(1/2)*(d*x^2+c)^(1/2)/a^(5/2)/(-a*d+b*
c)^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {425, 541, 539, 429, 422} \begin {gather*} \frac {4 b x \sqrt {c+d x^2} (b c-2 a d)}{15 a^2 \left (a+b x^2\right )^{3/2} (b c-a d)^2}+\frac {\sqrt {b} \sqrt {c+d x^2} \left (23 a^2 d^2-23 a b c d+8 b^2 c^2\right ) E\left (\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} \sqrt {a+b x^2} (b c-a d)^3 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \left (15 a^2 d^2-11 a b c d+4 b^2 c^2\right ) F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x \sqrt {c+d x^2}}{5 a \left (a+b x^2\right )^{5/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[c + d*x^2])/(5*a*(b*c - a*d)*(a + b*x^2)^(5/2)) + (4*b*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(15*a^2*(b*c
- a*d)^2*(a + b*x^2)^(3/2)) + (Sqrt[b]*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*Sqrt[c + d*x^2]*EllipticE[ArcTan
[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(15*a^(5/2)*(b*c - a*d)^3*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a
+ b*x^2))]) - (Sqrt[c]*Sqrt[d]*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]
*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*a^3*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[Sqrt[a + b
*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] && PosQ[d/c]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}} \, dx &=\frac {b x \sqrt {c+d x^2}}{5 a (b c-a d) \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {-4 b c+5 a d-3 b d x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx}{5 a (b c-a d)}\\ &=\frac {b x \sqrt {c+d x^2}}{5 a (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 b (b c-2 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {\int \frac {8 b^2 c^2-19 a b c d+15 a^2 d^2+4 b d (b c-2 a d) x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{15 a^2 (b c-a d)^2}\\ &=\frac {b x \sqrt {c+d x^2}}{5 a (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 b (b c-2 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}-\frac {\left (d \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 (b c-a d)^3}+\frac {\left (b \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^2 (b c-a d)^3}\\ &=\frac {b x \sqrt {c+d x^2}}{5 a (b c-a d) \left (a+b x^2\right )^{5/2}}+\frac {4 b (b c-2 a d) x \sqrt {c+d x^2}}{15 a^2 (b c-a d)^2 \left (a+b x^2\right )^{3/2}}+\frac {\sqrt {b} \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} (b c-a d)^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {d} \left (4 b^2 c^2-11 a b c d+15 a^2 d^2\right ) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^3 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 3.55, size = 301, normalized size = 0.90 \begin {gather*} \frac {b \sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (3 a^2 (b c-a d)^2+4 a (b c-2 a d) (b c-a d) \left (a+b x^2\right )+\left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) \left (a+b x^2\right )^2\right )+i \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (b c \left (8 b^2 c^2-23 a b c d+23 a^2 d^2\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+\left (-8 b^3 c^3+27 a b^2 c^2 d-34 a^2 b c d^2+15 a^3 d^3\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )\right )}{15 a^3 \sqrt {\frac {b}{a}} (b c-a d)^3 \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[b/a]*x*(c + d*x^2)*(3*a^2*(b*c - a*d)^2 + 4*a*(b*c - 2*a*d)*(b*c - a*d)*(a + b*x^2) + (8*b^2*c^2 - 23*
a*b*c*d + 23*a^2*d^2)*(a + b*x^2)^2) + I*(a + b*x^2)^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(b*c*(8*b^2*c^2
- 23*a*b*c*d + 23*a^2*d^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (-8*b^3*c^3 + 27*a*b^2*c^2*d - 34
*a^2*b*c*d^2 + 15*a^3*d^3)*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(15*a^3*Sqrt[b/a]*(b*c - a*d)^3*(a
+ b*x^2)^(5/2)*Sqrt[c + d*x^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1606\) vs. \(2(368)=736\).
time = 0.08, size = 1607, normalized size = 4.81

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{5 b^{2} a \left (a d -b c \right ) \left (x^{2}+\frac {a}{b}\right )^{3}}-\frac {4 \left (2 a d -b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}{15 b \,a^{2} \left (a d -b c \right )^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {\left (b d \,x^{2}+b c \right ) x \left (23 a^{2} d^{2}-23 a b c d +8 b^{2} c^{2}\right )}{15 a^{3} \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {4 d \left (2 a d -b c \right )}{15 \left (a d -b c \right )^{2} a^{2}}+\frac {23 a^{2} d^{2}-23 a b c d +8 b^{2} c^{2}}{15 \left (a d -b c \right )^{2} a^{3}}+\frac {b c \left (23 a^{2} d^{2}-23 a b c d +8 b^{2} c^{2}\right )}{15 a^{3} \left (a d -b c \right )^{3}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}-\frac {b \left (23 a^{2} d^{2}-23 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\EllipticF \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\EllipticE \left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{15 a^{3} \left (a d -b c \right )^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c \,x^{2} b +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(573\)
default	\(\text {Expression too large to display}\)	\(1607\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(15*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b^2*d^3*x^4+30*
((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^4*b*d^3*x^2-16*((b*x^2+a)/
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^4*c^3*x^2+16*((b*x^2+a)/a)^(1/2)*((
d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^4*c^3*x^2-34*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)
^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^4*b*c*d^2+27*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipt
icF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b^2*c^2*d+23*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a
)^(1/2),(a*d/b/c)^(1/2))*a^4*b*c*d^2-23*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/
b/c)^(1/2))*a^3*b^2*c^2*d-8*(-b/a)^(1/2)*b^5*c^3*x^5+54*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-
b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c^2*d*x^2+15*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1
/2),(a*d/b/c)^(1/2))*a^5*d^3-23*(-b/a)^(1/2)*a^2*b^3*d^3*x^7-8*(-b/a)^(1/2)*b^5*c^2*d*x^7-54*(-b/a)^(1/2)*a^3*
b^2*d^3*x^5-34*(-b/a)^(1/2)*a^4*b*d^3*x^3-20*(-b/a)^(1/2)*a*b^4*c^3*x^3-15*(-b/a)^(1/2)*a^2*b^3*c^3*x-34*((b*x
^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c*d^2*x^4+27*((b*x^2+a)/a
)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^4*c^2*d*x^4+23*((b*x^2+a)/a)^(1/2)*(
(d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c*d^2*x^4-23*((b*x^2+a)/a)^(1/2)*((d*x^2+
c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b^4*c^2*d*x^4-68*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/
2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b^2*c*d^2*x^2+46*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli
pticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^3*b^2*c*d^2*x^2-46*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x
*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c^2*d*x^2+23*(-b/a)^(1/2)*a*b^4*c*d^2*x^7+35*(-b/a)^(1/2)*a^2*b^3*c*d^2
*x^5+3*(-b/a)^(1/2)*a*b^4*c^2*d*x^5-13*(-b/a)^(1/2)*a^3*b^2*c*d^2*x^3+43*(-b/a)^(1/2)*a^2*b^3*c^2*d*x^3-34*(-b
/a)^(1/2)*a^4*b*c*d^2*x+41*(-b/a)^(1/2)*a^3*b^2*c^2*d*x-8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*
(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^5*c^3*x^4+8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(
a*d/b/c)^(1/2))*b^5*c^3*x^4-8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)
)*a^2*b^3*c^3+8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*b^3*c^3)
/(d*x^2+c)^(1/2)/(a*d-b*c)^3/a^3/(-b/a)^(1/2)/(b*x^2+a)^(5/2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x^{2}\right )^{\frac {7}{2}} \sqrt {c + d x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x**2)**(7/2)*sqrt(c + d*x**2)), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________