\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^4} \, dx\) [60]

Optimal result

Integrand size = 21, antiderivative size = 199 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {a^2 (6 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}} \]

[Out]

-1/6*d*x*(b*x^2+a)^(5/2)/c/(-a*d+b*c)/(d*x^2+c)^3+1/24*(-5*a*d+6*b*c)*x*(b*x^2+a)^(3/2)/c^2/(-a*d+b*c)/(d*x^2+
c)^2+1/16*a^2*(-5*a*d+6*b*c)*arctanh(x*(-a*d+b*c)^(1/2)/c^(1/2)/(b*x^2+a)^(1/2))/c^(7/2)/(-a*d+b*c)^(3/2)+1/16
*a*(-5*a*d+6*b*c)*x*(b*x^2+a)^(1/2)/c^3/(-a*d+b*c)/(d*x^2+c)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {390, 386, 385, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\frac {a^2 (6 b c-5 a d) \text {arctanh}\left (\frac {x \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}}+\frac {a x \sqrt {a+b x^2} (6 b c-5 a d)}{16 c^3 \left (c+d x^2\right ) (b c-a d)}+\frac {x \left (a+b x^2\right )^{3/2} (6 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac {d x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3 (b c-a d)} \]

[In]

Int[(a + b*x^2)^(3/2)/(c + d*x^2)^4,x]

[Out]

-1/6*(d*x*(a + b*x^2)^(5/2))/(c*(b*c - a*d)*(c + d*x^2)^3) + ((6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2))/(24*c^2*(b*
c - a*d)*(c + d*x^2)^2) + (a*(6*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(16*c^3*(b*c - a*d)*(c + d*x^2)) + (a^2*(6*b*c
- 5*a*d)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(3/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 386

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

Rule 390

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[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx}{6 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {(a (6 b c-5 a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^2} \, dx}{8 c^2 (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {\left (a^2 (6 b c-5 a d)\right ) \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )} \, dx}{16 c^3 (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {\left (a^2 (6 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{c-(b c-a d) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 c^3 (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{5/2}}{6 c (b c-a d) \left (c+d x^2\right )^3}+\frac {(6 b c-5 a d) x \left (a+b x^2\right )^{3/2}}{24 c^2 (b c-a d) \left (c+d x^2\right )^2}+\frac {a (6 b c-5 a d) x \sqrt {a+b x^2}}{16 c^3 (b c-a d) \left (c+d x^2\right )}+\frac {a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.51 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\frac {a x \left (1+\frac {b x^2}{a}\right ) \left (c \left (a+b x^2\right ) \left (-4 b^2 c^2 x^2 \left (3 c+d x^2\right )-2 a b c \left (15 c^2+11 c d x^2+4 d^2 x^4\right )+a^2 d \left (33 c^2+40 c d x^2+15 d^2 x^4\right )\right )+\frac {3 a^2 (-6 b c+5 a d) \left (c+d x^2\right )^3 \text {arctanh}\left (\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}\right )}{\sqrt {\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}}}\right )}{48 c^4 (-b c+a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^3} \]

[In]

Integrate[(a + b*x^2)^(3/2)/(c + d*x^2)^4,x]

[Out]

(a*x*(1 + (b*x^2)/a)*(c*(a + b*x^2)*(-4*b^2*c^2*x^2*(3*c + d*x^2) - 2*a*b*c*(15*c^2 + 11*c*d*x^2 + 4*d^2*x^4)
+ a^2*d*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)) + (3*a^2*(-6*b*c + 5*a*d)*(c + d*x^2)^3*ArcTanh[Sqrt[((b*c - a*d)*
x^2)/(c*(a + b*x^2))]])/Sqrt[((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(48*c^4*(-(b*c) + a*d)*(a + b*x^2)^(3/2)*(c
+ d*x^2)^3)

Maple [A] (verified)

Time = 2.58 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {-11 x \sqrt {b \,x^{2}+a}\, \left (-\frac {10 \left (\frac {2 b \,x^{2}}{5}+a \right ) b \,c^{3}}{11}+d \left (-\frac {4}{33} b^{2} x^{4}-\frac {2}{3} a b \,x^{2}+a^{2}\right ) c^{2}+\frac {40 x^{2} \left (-\frac {b \,x^{2}}{5}+a \right ) d^{2} a c}{33}+\frac {5 a^{2} d^{3} x^{4}}{11}\right ) \sqrt {\left (a d -b c \right ) c}+5 \left (d \,x^{2}+c \right )^{3} \left (a d -\frac {6 b c}{5}\right ) a^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{16 \sqrt {\left (a d -b c \right ) c}\, \left (a d -b c \right ) c^{3} \left (d \,x^{2}+c \right )^{3}}\)	\(179\)
default	\(\text {Expression too large to display}\)	\(12816\)

[In]

int((b*x^2+a)^(3/2)/(d*x^2+c)^4,x,method=_RETURNVERBOSE)

[Out]

-1/16/((a*d-b*c)*c)^(1/2)*(-11*x*(b*x^2+a)^(1/2)*(-10/11*(2/5*b*x^2+a)*b*c^3+d*(-4/33*b^2*x^4-2/3*a*b*x^2+a^2)
*c^2+40/33*x^2*(-1/5*b*x^2+a)*d^2*a*c+5/11*a^2*d^3*x^4)*((a*d-b*c)*c)^(1/2)+5*(d*x^2+c)^3*(a*d-6/5*b*c)*a^2*ar
ctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2)))/(a*d-b*c)/c^3/(d*x^2+c)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (175) = 350\).

Time = 0.44 (sec) , antiderivative size = 972, normalized size of antiderivative = 4.88 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\left [\frac {3 \, {\left (6 \, a^{2} b c^{4} - 5 \, a^{3} c^{3} d + {\left (6 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{6} + 3 \, {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{4} + 3 \, {\left (6 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (4 \, b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 23 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{5} + 2 \, {\left (6 \, b^{3} c^{5} + 5 \, a b^{2} c^{4} d - 31 \, a^{2} b c^{3} d^{2} + 20 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c^{5} - 21 \, a^{2} b c^{4} d + 11 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{192 \, {\left (b^{2} c^{9} - 2 \, a b c^{8} d + a^{2} c^{7} d^{2} + {\left (b^{2} c^{6} d^{3} - 2 \, a b c^{5} d^{4} + a^{2} c^{4} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{7} d^{2} - 2 \, a b c^{6} d^{3} + a^{2} c^{5} d^{4}\right )} x^{4} + 3 \, {\left (b^{2} c^{8} d - 2 \, a b c^{7} d^{2} + a^{2} c^{6} d^{3}\right )} x^{2}\right )}}, -\frac {3 \, {\left (6 \, a^{2} b c^{4} - 5 \, a^{3} c^{3} d + {\left (6 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{6} + 3 \, {\left (6 \, a^{2} b c^{2} d^{2} - 5 \, a^{3} c d^{3}\right )} x^{4} + 3 \, {\left (6 \, a^{2} b c^{3} d - 5 \, a^{3} c^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (4 \, b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 23 \, a^{2} b c^{2} d^{3} + 15 \, a^{3} c d^{4}\right )} x^{5} + 2 \, {\left (6 \, b^{3} c^{5} + 5 \, a b^{2} c^{4} d - 31 \, a^{2} b c^{3} d^{2} + 20 \, a^{3} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (10 \, a b^{2} c^{5} - 21 \, a^{2} b c^{4} d + 11 \, a^{3} c^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{96 \, {\left (b^{2} c^{9} - 2 \, a b c^{8} d + a^{2} c^{7} d^{2} + {\left (b^{2} c^{6} d^{3} - 2 \, a b c^{5} d^{4} + a^{2} c^{4} d^{5}\right )} x^{6} + 3 \, {\left (b^{2} c^{7} d^{2} - 2 \, a b c^{6} d^{3} + a^{2} c^{5} d^{4}\right )} x^{4} + 3 \, {\left (b^{2} c^{8} d - 2 \, a b c^{7} d^{2} + a^{2} c^{6} d^{3}\right )} x^{2}\right )}}\right ] \]

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^4,x, algorithm="fricas")

[Out]

[1/192*(3*(6*a^2*b*c^4 - 5*a^3*c^3*d + (6*a^2*b*c*d^3 - 5*a^3*d^4)*x^6 + 3*(6*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x^4
+ 3*(6*a^2*b*c^3*d - 5*a^3*c^2*d^2)*x^2)*sqrt(b*c^2 - a*c*d)*log(((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2
*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2 + 4*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a))/(d^2
*x^4 + 2*c*d*x^2 + c^2)) + 4*((4*b^3*c^4*d + 4*a*b^2*c^3*d^2 - 23*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^5 + 2*(6*b^3
*c^5 + 5*a*b^2*c^4*d - 31*a^2*b*c^3*d^2 + 20*a^3*c^2*d^3)*x^3 + 3*(10*a*b^2*c^5 - 21*a^2*b*c^4*d + 11*a^3*c^3*
d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^9 - 2*a*b*c^8*d + a^2*c^7*d^2 + (b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x^
6 + 3*(b^2*c^7*d^2 - 2*a*b*c^6*d^3 + a^2*c^5*d^4)*x^4 + 3*(b^2*c^8*d - 2*a*b*c^7*d^2 + a^2*c^6*d^3)*x^2), -1/9
6*(3*(6*a^2*b*c^4 - 5*a^3*c^3*d + (6*a^2*b*c*d^3 - 5*a^3*d^4)*x^6 + 3*(6*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x^4 + 3*
(6*a^2*b*c^3*d - 5*a^3*c^2*d^2)*x^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 +
a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 2*((4*b^3*c^4*d + 4*a*b^2*c^3*d^2 -
23*a^2*b*c^2*d^3 + 15*a^3*c*d^4)*x^5 + 2*(6*b^3*c^5 + 5*a*b^2*c^4*d - 31*a^2*b*c^3*d^2 + 20*a^3*c^2*d^3)*x^3
+ 3*(10*a*b^2*c^5 - 21*a^2*b*c^4*d + 11*a^3*c^3*d^2)*x)*sqrt(b*x^2 + a))/(b^2*c^9 - 2*a*b*c^8*d + a^2*c^7*d^2
+ (b^2*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x^6 + 3*(b^2*c^7*d^2 - 2*a*b*c^6*d^3 + a^2*c^5*d^4)*x^4 + 3*(b^2
*c^8*d - 2*a*b*c^7*d^2 + a^2*c^6*d^3)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\text {Timed out} \]

[In]

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{4}} \,d x } \]

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^4,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^4, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (175) = 350\).

Time = 1.34 (sec) , antiderivative size = 919, normalized size of antiderivative = 4.62 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=-\frac {{\left (6 \, a^{2} b^{\frac {3}{2}} c - 5 \, a^{3} \sqrt {b} d\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{16 \, {\left (b c^{4} - a c^{3} d\right )} \sqrt {-b^{2} c^{2} + a b c d}} - \frac {18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {3}{2}} c d^{4} - 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} \sqrt {b} d^{5} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c^{4} d + 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} c^{3} d^{2} + 180 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} c^{2} d^{3} - 240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} c d^{4} + 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} \sqrt {b} d^{5} - 128 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {11}{2}} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c^{4} d + 720 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} c^{3} d^{2} - 968 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} c^{2} d^{3} + 620 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} c d^{4} - 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} \sqrt {b} d^{5} - 96 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c^{4} d - 288 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} c^{3} d^{2} + 864 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} c^{2} d^{3} - 600 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} c d^{4} + 150 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} \sqrt {b} d^{5} - 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} c^{3} d^{2} - 72 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} c^{2} d^{3} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} c d^{4} - 75 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} \sqrt {b} d^{5} - 4 \, a^{6} b^{\frac {5}{2}} c^{2} d^{3} - 8 \, a^{7} b^{\frac {3}{2}} c d^{4} + 15 \, a^{8} \sqrt {b} d^{5}}{24 \, {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{3}} \]

[In]

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^4,x, algorithm="giac")

[Out]

-1/16*(6*a^2*b^(3/2)*c - 5*a^3*sqrt(b)*d)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a*d)/sqrt(-b
^2*c^2 + a*b*c*d))/((b*c^4 - a*c^3*d)*sqrt(-b^2*c^2 + a*b*c*d)) - 1/24*(18*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^
2*b^(3/2)*c*d^4 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*sqrt(b)*d^5 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(
9/2)*c^4*d + 96*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*c^3*d^2 + 180*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^
(5/2)*c^2*d^3 - 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*c*d^4 + 75*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^4
*sqrt(b)*d^5 - 128*(sqrt(b)*x - sqrt(b*x^2 + a))^6*b^(11/2)*c^5 - 64*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(9/2)
*c^4*d + 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(7/2)*c^3*d^2 - 968*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^(
5/2)*c^2*d^3 + 620*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(3/2)*c*d^4 - 150*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5
*sqrt(b)*d^5 - 96*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^2*b^(9/2)*c^4*d - 288*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*
b^(7/2)*c^3*d^2 + 864*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*c^2*d^3 - 600*(sqrt(b)*x - sqrt(b*x^2 + a))^
4*a^5*b^(3/2)*c*d^4 + 150*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^6*sqrt(b)*d^5 - 48*(sqrt(b)*x - sqrt(b*x^2 + a))^2
*a^4*b^(7/2)*c^3*d^2 - 72*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*c^2*d^3 + 210*(sqrt(b)*x - sqrt(b*x^2 +
a))^2*a^6*b^(3/2)*c*d^4 - 75*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*sqrt(b)*d^5 - 4*a^6*b^(5/2)*c^2*d^3 - 8*a^7*b
^(3/2)*c*d^4 + 15*a^8*sqrt(b)*d^5)/((b*c^4*d^2 - a*c^3*d^3)*((sqrt(b)*x - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x
- sqrt(b*x^2 + a))^2*b*c - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^4} \,d x \]

[In]

int((a + b*x^2)^(3/2)/(c + d*x^2)^4,x)

[Out]

int((a + b*x^2)^(3/2)/(c + d*x^2)^4, x)