\(\int \frac {x^5}{(a+b x^2)^{9/2} \sqrt {c+d x^2}} \, dx\) [989]

Optimal result

Integrand size = 26, antiderivative size = 217 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {2 a (7 b c-4 a d) \sqrt {c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac {\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}+\frac {2 d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^4 \sqrt {a+b x^2}} \]

[Out]

-1/7*a^2*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)/(b*x^2+a)^(7/2)+2/35*a*(-4*a*d+7*b*c)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^2
/(b*x^2+a)^(5/2)-1/105*(3*a^2*d^2-14*a*b*c*d+35*b^2*c^2)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^3/(b*x^2+a)^(3/2)+2/10
5*d*(3*a^2*d^2-14*a*b*c*d+35*b^2*c^2)*(d*x^2+c)^(1/2)/b^2/(-a*d+b*c)^4/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {457, 91, 79, 47, 37} \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {2 d \sqrt {c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt {a+b x^2} (b c-a d)^4}-\frac {\sqrt {c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac {a^2 \sqrt {c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac {2 a \sqrt {c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]

[In]

Int[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

-1/7*(a^2*Sqrt[c + d*x^2])/(b^2*(b*c - a*d)*(a + b*x^2)^(7/2)) + (2*a*(7*b*c - 4*a*d)*Sqrt[c + d*x^2])/(35*b^2
*(b*c - a*d)^2*(a + b*x^2)^(5/2)) - ((35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c - a*
d)^3*(a + b*x^2)^(3/2)) + (2*d*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Sqrt[c + d*x^2])/(105*b^2*(b*c - a*d)^4*S
qrt[a + b*x^2])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 457

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{2} a (7 b c-a d)+\frac {7}{2} b (b c-a d) x}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{7 b^2 (b c-a d)} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {2 a (7 b c-4 a d) \sqrt {c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}+\frac {\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{70 b^2 (b c-a d)^2} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {2 a (7 b c-4 a d) \sqrt {c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac {\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx,x,x^2\right )}{105 b^2 (b c-a d)^3} \\ & = -\frac {a^2 \sqrt {c+d x^2}}{7 b^2 (b c-a d) \left (a+b x^2\right )^{7/2}}+\frac {2 a (7 b c-4 a d) \sqrt {c+d x^2}}{35 b^2 (b c-a d)^2 \left (a+b x^2\right )^{5/2}}-\frac {\left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^3 \left (a+b x^2\right )^{3/2}}+\frac {2 d \left (35 b^2 c^2-14 a b c d+3 a^2 d^2\right ) \sqrt {c+d x^2}}{105 b^2 (b c-a d)^4 \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.49 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-35 b^3 c^2 x^4 \left (c-2 d x^2\right )+7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )+a^2 b \left (-8 c^3+200 c^2 d x^2-101 c d^2 x^4+6 d^3 x^6\right )\right )}{105 (b c-a d)^4 \left (a+b x^2\right )^{7/2}} \]

[In]

Integrate[x^5/((a + b*x^2)^(9/2)*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[c + d*x^2]*(-35*b^3*c^2*x^4*(c - 2*d*x^2) + 7*a^3*d*(8*c^2 - 4*c*d*x^2 + 3*d^2*x^4) - 7*a*b^2*c*x^2*(4*c
^2 - 37*c*d*x^2 + 4*d^2*x^4) + a^2*b*(-8*c^3 + 200*c^2*d*x^2 - 101*c*d^2*x^4 + 6*d^3*x^6)))/(105*(b*c - a*d)^4
*(a + b*x^2)^(7/2))

Maple [A] (verified)

Time = 3.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}\)	\(205\)
gosper	\(\frac {\sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(213\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {d \,x^{2}+c}\, \left (6 a^{2} b \,d^{3} x^{6}-28 a \,b^{2} c \,d^{2} x^{6}+70 b^{3} c^{2} d \,x^{6}+21 a^{3} d^{3} x^{4}-101 a^{2} b c \,d^{2} x^{4}+259 a \,b^{2} c^{2} d \,x^{4}-35 c^{3} b^{3} x^{4}-28 a^{3} c \,d^{2} x^{2}+200 a^{2} b \,c^{2} d \,x^{2}-28 a \,b^{2} c^{3} x^{2}+56 a^{3} c^{2} d -8 a^{2} b \,c^{3}\right )}{105 \sqrt {b \,x^{2}+a}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, \left (b^{3} x^{6}+3 a \,b^{2} x^{4}+3 a^{2} b \,x^{2}+a^{3}\right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\)	\(285\)

[In]

int(x^5/(b*x^2+a)^(9/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/105*(d*x^2+c)^(1/2)*(6*a^2*b*d^3*x^6-28*a*b^2*c*d^2*x^6+70*b^3*c^2*d*x^6+21*a^3*d^3*x^4-101*a^2*b*c*d^2*x^4+
259*a*b^2*c^2*d*x^4-35*b^3*c^3*x^4-28*a^3*c*d^2*x^2+200*a^2*b*c^2*d*x^2-28*a*b^2*c^3*x^2+56*a^3*c^2*d-8*a^2*b*
c^3)/(b*x^2+a)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (193) = 386\).

Time = 0.69 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.08 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {{\left (2 \, {\left (35 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{6} - 8 \, a^{2} b c^{3} + 56 \, a^{3} c^{2} d - {\left (35 \, b^{3} c^{3} - 259 \, a b^{2} c^{2} d + 101 \, a^{2} b c d^{2} - 21 \, a^{3} d^{3}\right )} x^{4} - 4 \, {\left (7 \, a b^{2} c^{3} - 50 \, a^{2} b c^{2} d + 7 \, a^{3} c d^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{8} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{6} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{4} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x^{2}\right )}} \]

[In]

integrate(x^5/(b*x^2+a)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

1/105*(2*(35*b^3*c^2*d - 14*a*b^2*c*d^2 + 3*a^2*b*d^3)*x^6 - 8*a^2*b*c^3 + 56*a^3*c^2*d - (35*b^3*c^3 - 259*a*
b^2*c^2*d + 101*a^2*b*c*d^2 - 21*a^3*d^3)*x^4 - 4*(7*a*b^2*c^3 - 50*a^2*b*c^2*d + 7*a^3*c*d^2)*x^2)*sqrt(b*x^2
+ a)*sqrt(d*x^2 + c)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2 - 4*a^7*b*c*d^3 + a^8*d^4 + (b^8*c^4
- 4*a*b^7*c^3*d + 6*a^2*b^6*c^2*d^2 - 4*a^3*b^5*c*d^3 + a^4*b^4*d^4)*x^8 + 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*
a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^6 + 6*(a^2*b^6*c^4 - 4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 -
4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^4 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d + 6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 +
a^7*b*d^4)*x^2)

Sympy [F]

\[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{2}\right )^{\frac {9}{2}} \sqrt {c + d x^{2}}}\, dx \]

[In]

integrate(x**5/(b*x**2+a)**(9/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(x**5/((a + b*x**2)**(9/2)*sqrt(c + d*x**2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^5/(b*x^2+a)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (193) = 386\).

Time = 0.42 (sec) , antiderivative size = 1036, normalized size of antiderivative = 4.77 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {4 \, {\left (35 \, \sqrt {b d} b^{10} c^{5} d - 119 \, \sqrt {b d} a b^{9} c^{4} d^{2} + 150 \, \sqrt {b d} a^{2} b^{8} c^{3} d^{3} - 86 \, \sqrt {b d} a^{3} b^{7} c^{2} d^{4} + 23 \, \sqrt {b d} a^{4} b^{6} c d^{5} - 3 \, \sqrt {b d} a^{5} b^{5} d^{6} - 245 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{8} c^{4} d + 588 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b^{7} c^{3} d^{2} - 462 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{2} b^{6} c^{2} d^{3} + 140 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{3} b^{5} c d^{4} - 21 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a^{4} b^{4} d^{5} + 630 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} b^{6} c^{3} d - 714 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a b^{5} c^{2} d^{2} + 42 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{2} b^{4} c d^{3} + 42 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4} a^{3} b^{3} d^{4} - 770 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} b^{4} c^{2} d + 140 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a b^{3} c d^{2} - 210 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{6} a^{2} b^{2} d^{3} + 455 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{8} b^{2} c d + 105 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{8} a b d^{2} - 105 \, \sqrt {b d} {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{10} d\right )}}{105 \, {\left (b^{2} c - a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}^{7} {\left | b \right |}} \]

[In]

integrate(x^5/(b*x^2+a)^(9/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

4/105*(35*sqrt(b*d)*b^10*c^5*d - 119*sqrt(b*d)*a*b^9*c^4*d^2 + 150*sqrt(b*d)*a^2*b^8*c^3*d^3 - 86*sqrt(b*d)*a^
3*b^7*c^2*d^4 + 23*sqrt(b*d)*a^4*b^6*c*d^5 - 3*sqrt(b*d)*a^5*b^5*d^6 - 245*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d
) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*b^8*c^4*d + 588*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c
+ (b*x^2 + a)*b*d - a*b*d))^2*a*b^7*c^3*d^2 - 462*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2
+ a)*b*d - a*b*d))^2*a^2*b^6*c^2*d^3 + 140*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d
- a*b*d))^2*a^3*b^5*c*d^4 - 21*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^
2*a^4*b^4*d^5 + 630*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*b^6*c^3*d
- 714*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a*b^5*c^2*d^2 + 42*sqrt(
b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a^2*b^4*c*d^3 + 42*sqrt(b*d)*(sqrt(
b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^4*a^3*b^3*d^4 - 770*sqrt(b*d)*(sqrt(b*x^2 + a)*s
qrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*b^4*c^2*d + 140*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqr
t(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*a*b^3*c*d^2 - 210*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b
*x^2 + a)*b*d - a*b*d))^6*a^2*b^2*d^3 + 455*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*
d - a*b*d))^8*b^2*c*d + 105*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^8*a*
b*d^2 - 105*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^10*d)/((b^2*c - a*b*
d - (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)^7*abs(b))

Mupad [B] (verification not implemented)

Time = 6.46 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.55 \[ \int \frac {x^5}{\left (a+b x^2\right )^{9/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b\,x^2+a}\,\left (\frac {x^6\,\left (21\,a^3\,d^4-95\,a^2\,b\,c\,d^3+231\,a\,b^2\,c^2\,d^2+35\,b^3\,c^3\,d\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}-\frac {x^4\,\left (7\,a^3\,c\,d^3-99\,a^2\,b\,c^2\,d^2-231\,a\,b^2\,c^3\,d+35\,b^3\,c^4\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {8\,a^2\,c^3\,\left (7\,a\,d-b\,c\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,d^2\,x^8\,\left (3\,a^2\,d^2-14\,a\,b\,c\,d+35\,b^2\,c^2\right )}{105\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,a\,c^2\,x^2\,\left (7\,a^2\,d^2+48\,a\,b\,c\,d-7\,b^2\,c^2\right )}{105\,b^4\,{\left (a\,d-b\,c\right )}^4}\right )}{x^8\,\sqrt {d\,x^2+c}+\frac {a^4\,\sqrt {d\,x^2+c}}{b^4}+\frac {4\,a\,x^6\,\sqrt {d\,x^2+c}}{b}+\frac {6\,a^2\,x^4\,\sqrt {d\,x^2+c}}{b^2}+\frac {4\,a^3\,x^2\,\sqrt {d\,x^2+c}}{b^3}} \]

[In]

int(x^5/((a + b*x^2)^(9/2)*(c + d*x^2)^(1/2)),x)

[Out]

((a + b*x^2)^(1/2)*((x^6*(21*a^3*d^4 + 35*b^3*c^3*d + 231*a*b^2*c^2*d^2 - 95*a^2*b*c*d^3))/(105*b^4*(a*d - b*c
)^4) - (x^4*(35*b^3*c^4 + 7*a^3*c*d^3 - 99*a^2*b*c^2*d^2 - 231*a*b^2*c^3*d))/(105*b^4*(a*d - b*c)^4) + (8*a^2*
c^3*(7*a*d - b*c))/(105*b^4*(a*d - b*c)^4) + (2*d^2*x^8*(3*a^2*d^2 + 35*b^2*c^2 - 14*a*b*c*d))/(105*b^3*(a*d -
b*c)^4) + (4*a*c^2*x^2*(7*a^2*d^2 - 7*b^2*c^2 + 48*a*b*c*d))/(105*b^4*(a*d - b*c)^4)))/(x^8*(c + d*x^2)^(1/2)
+ (a^4*(c + d*x^2)^(1/2))/b^4 + (4*a*x^6*(c + d*x^2)^(1/2))/b + (6*a^2*x^4*(c + d*x^2)^(1/2))/b^2 + (4*a^3*x^
2*(c + d*x^2)^(1/2))/b^3)