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

Optimal. Leaf size=217 \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(7*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*Sqr
t[a + b*x^2])

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Rubi [A]  time = 0.268655, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {446, 89, 78, 45, 37} \[ \frac{2 d \sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \sqrt{a+b x^2} (b c-a d)^4}-\frac{\sqrt{c+d x^2} \left (3 a^2 d^2-14 a b c d+35 b^2 c^2\right )}{105 b^2 \left (a+b x^2\right )^{3/2} (b c-a d)^3}-\frac{a^2 \sqrt{c+d x^2}}{7 b^2 \left (a+b x^2\right )^{7/2} (b c-a d)}+\frac{2 a \sqrt{c+d x^2} (7 b c-4 a d)}{35 b^2 \left (a+b x^2\right )^{5/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*Sqrt[c + d*x^2])/(7*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*Sqr
t[a + b*x^2])

Rule 446

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]]

Rule 89

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 78

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] || LtQ
[p, n]))))

Rule 45

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 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]

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{9/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.0763455, size = 151, normalized size = 0.7 \[ \frac{\sqrt{c+d x^2} \left (a^2 b \left (200 c^2 d x^2-8 c^3-101 c d^2 x^4+6 d^3 x^6\right )+7 a^3 d \left (8 c^2-4 c d x^2+3 d^2 x^4\right )-7 a b^2 c x^2 \left (4 c^2-37 c d x^2+4 d^2 x^4\right )-35 b^3 c^2 x^4 \left (c-2 d x^2\right )\right )}{105 \left (a+b x^2\right )^{7/2} (b c-a d)^4} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.01, size = 213, normalized size = 1. \begin{align*}{\frac{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}bc{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}}{105\,{a}^{4}{d}^{4}-420\,{a}^{3}bc{d}^{3}+630\,{a}^{2}{c}^{2}{d}^{2}{b}^{2}-420\,a{c}^{3}d{b}^{3}+105\,{c}^{4}{b}^{4}}\sqrt{d{x}^{2}+c} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)^(7/2)/(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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.54495, size = 914, normalized size = 4.21 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.48882, size = 1407, normalized size = 6.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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^11*c^5*d - 119*sqrt(b*d)*a*b^10*c^4*d^2 + 150*sqrt(b*d)*a^2*b^9*c^3*d^3 - 86*sqrt(b*d)*a
^3*b^8*c^2*d^4 + 23*sqrt(b*d)*a^4*b^7*c*d^5 - 3*sqrt(b*d)*a^5*b^6*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^9*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^8*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^7*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^6*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^5*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^7*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^6*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^5*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^4*d^4 - 770*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*b^5*c^2*d + 140*sqrt(b*d)*(sqrt(b*x^2 + a)*sqrt(b*d) - sq
rt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^6*a*b^4*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^3*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^3*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^2*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*b*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*b*abs(b))