3.6.61 \(\int \frac {1}{x^4 (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\)

Optimal. Leaf size=149 \[ -\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{5/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^6} (3 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac {b \sqrt {c+d x^6}}{6 a x^3 \left (a+b x^6\right ) (b c-a d)} \]

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Rubi [A]  time = 0.19, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {465, 472, 583, 12, 377, 205} \begin {gather*} -\frac {\sqrt {c+d x^6} (3 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {x^3 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{5/2} (b c-a d)^{3/2}}+\frac {b \sqrt {c+d x^6}}{6 a x^3 \left (a+b x^6\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-((3*b*c - 2*a*d)*Sqrt[c + d*x^6])/(6*a^2*c*(b*c - a*d)*x^3) + (b*Sqrt[c + d*x^6])/(6*a*(b*c - a*d)*x^3*(a + b
*x^6)) - (b*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x^3)/(Sqrt[a]*Sqrt[c + d*x^6])])/(6*a^(5/2)*(b*c - a*d)^(3
/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 205

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

Rule 377

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 465

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

Rule 472

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

Rule 583

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

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx,x,x^3\right )\\ &=\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^3 \left (a+b x^6\right )}-\frac {\operatorname {Subst}\left (\int \frac {-3 b c+2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 a (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^6}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^3 \left (a+b x^6\right )}-\frac {\operatorname {Subst}\left (\int \frac {b c (3 b c-4 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 a^2 c (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^6}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^3 \left (a+b x^6\right )}-\frac {(b (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx,x,x^3\right )}{6 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^6}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^3 \left (a+b x^6\right )}-\frac {(b (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x^3}{\sqrt {c+d x^6}}\right )}{6 a^2 (b c-a d)}\\ &=-\frac {(3 b c-2 a d) \sqrt {c+d x^6}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^6}}{6 a (b c-a d) x^3 \left (a+b x^6\right )}-\frac {b (3 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x^3}{\sqrt {a} \sqrt {c+d x^6}}\right )}{6 a^{5/2} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [C]  time = 5.40, size = 869, normalized size = 5.83 \begin {gather*} -\frac {\sqrt {d x^6+c} \left (120 d^2 \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}}\right ) x^{12}+96 d^2 \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right ) x^{12}+32 d^2 \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right ) x^{12}-120 d^2 \sqrt {\frac {a (b c-a d) x^6 \left (d x^6+c\right )}{c^2 \left (b x^6+a\right )^2}} x^{12}+180 c d \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}}\right ) x^6+160 c d \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right ) x^6+64 c d \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right ) x^6-180 c d \sqrt {\frac {a (b c-a d) x^6 \left (d x^6+c\right )}{c^2 \left (b x^6+a\right )^2}} x^6+45 c^2 \sin ^{-1}\left (\sqrt {\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}}\right )+64 c^2 \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _2F_1\left (2,3;\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )+32 c^2 \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{5/2} \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}} \, _3F_2\left (2,2,3;1,\frac {7}{2};\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )-45 c^2 \sqrt {\frac {a (b c-a d) x^6 \left (d x^6+c\right )}{c^2 \left (b x^6+a\right )^2}}\right )}{90 c^3 x^3 \left (\frac {(b c-a d) x^6}{c \left (b x^6+a\right )}\right )^{3/2} \left (b x^6+a\right )^2 \sqrt {\frac {a \left (d x^6+c\right )}{c \left (b x^6+a\right )}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^4*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

-1/90*(Sqrt[c + d*x^6]*(-45*c^2*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] - 180*c*d*x^6*Sqrt[(
a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(a + b*x^6)^2)] - 120*d^2*x^12*Sqrt[(a*(b*c - a*d)*x^6*(c + d*x^6))/(c^2*(
a + b*x^6)^2)] + 45*c^2*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 180*c*d*x^6*ArcSin[Sqrt[((b*c - a*d)
*x^6)/(c*(a + b*x^6))]] + 120*d^2*x^12*ArcSin[Sqrt[((b*c - a*d)*x^6)/(c*(a + b*x^6))]] + 64*c^2*(((b*c - a*d)*
x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^
6)/(c*(a + b*x^6))] + 160*c*d*x^6*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6
))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 96*d^2*x^12*(((b*c - a*d)*x^6)/(c*(a + b
*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeometric2F1[2, 3, 7/2, ((b*c - a*d)*x^6)/(c*(a + b*x
^6))] + 32*c^2*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*HypergeometricP
FQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))] + 64*c*d*x^6*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5
/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*HypergeometricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x
^6))] + 32*d^2*x^12*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))^(5/2)*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))]*Hypergeome
tricPFQ[{2, 2, 3}, {1, 7/2}, ((b*c - a*d)*x^6)/(c*(a + b*x^6))]))/(c^3*x^3*(((b*c - a*d)*x^6)/(c*(a + b*x^6)))
^(3/2)*(a + b*x^6)^2*Sqrt[(a*(c + d*x^6))/(c*(a + b*x^6))])

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IntegrateAlgebraic [A]  time = 1.63, size = 159, normalized size = 1.07 \begin {gather*} \frac {\left (4 a b d-3 b^2 c\right ) \tan ^{-1}\left (\frac {a \sqrt {d}+b x^3 \sqrt {c+d x^6}+b \sqrt {d} x^6}{\sqrt {a} \sqrt {b c-a d}}\right )}{6 a^{5/2} (b c-a d)^{3/2}}+\frac {\sqrt {c+d x^6} \left (-2 a^2 d+2 a b c-2 a b d x^6+3 b^2 c x^6\right )}{6 a^2 c x^3 \left (a+b x^6\right ) (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^4*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]

[Out]

(Sqrt[c + d*x^6]*(2*a*b*c - 2*a^2*d + 3*b^2*c*x^6 - 2*a*b*d*x^6))/(6*a^2*c*(-(b*c) + a*d)*x^3*(a + b*x^6)) + (
(-3*b^2*c + 4*a*b*d)*ArcTan[(a*Sqrt[d] + b*Sqrt[d]*x^6 + b*x^3*Sqrt[c + d*x^6])/(Sqrt[a]*Sqrt[b*c - a*d])])/(6
*a^(5/2)*(b*c - a*d)^(3/2))

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fricas [B]  time = 0.61, size = 612, normalized size = 4.11 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{9} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{12} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{6} + a^{2} c^{2} + 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{9} - a c x^{3}\right )} \sqrt {d x^{6} + c} \sqrt {-a b c + a^{2} d}}{b^{2} x^{12} + 2 \, a b x^{6} + a^{2}}\right ) + 4 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{6} + 2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2}\right )} \sqrt {d x^{6} + c}}{24 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{9} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3}\right )}}, -\frac {{\left ({\left (3 \, b^{3} c^{2} - 4 \, a b^{2} c d\right )} x^{9} + {\left (3 \, a b^{2} c^{2} - 4 \, a^{2} b c d\right )} x^{3}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c\right )} \sqrt {d x^{6} + c} \sqrt {a b c - a^{2} d}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{9} + {\left (a b c^{2} - a^{2} c d\right )} x^{3}\right )}}\right ) + 2 \, {\left ({\left (3 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{6} + 2 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + 2 \, a^{4} d^{2}\right )} \sqrt {d x^{6} + c}}{12 \, {\left ({\left (a^{3} b^{3} c^{3} - 2 \, a^{4} b^{2} c^{2} d + a^{5} b c d^{2}\right )} x^{9} + {\left (a^{4} b^{2} c^{3} - 2 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")

[Out]

[-1/24*(((3*b^3*c^2 - 4*a*b^2*c*d)*x^9 + (3*a*b^2*c^2 - 4*a^2*b*c*d)*x^3)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 -
8*a*b*c*d + 8*a^2*d^2)*x^12 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^6 + a^2*c^2 + 4*((b*c - 2*a*d)*x^9 - a*c*x^3)*sqrt(
d*x^6 + c)*sqrt(-a*b*c + a^2*d))/(b^2*x^12 + 2*a*b*x^6 + a^2)) + 4*((3*a*b^3*c^2 - 5*a^2*b^2*c*d + 2*a^3*b*d^2
)*x^6 + 2*a^2*b^2*c^2 - 4*a^3*b*c*d + 2*a^4*d^2)*sqrt(d*x^6 + c))/((a^3*b^3*c^3 - 2*a^4*b^2*c^2*d + a^5*b*c*d^
2)*x^9 + (a^4*b^2*c^3 - 2*a^5*b*c^2*d + a^6*c*d^2)*x^3), -1/12*(((3*b^3*c^2 - 4*a*b^2*c*d)*x^9 + (3*a*b^2*c^2
- 4*a^2*b*c*d)*x^3)*sqrt(a*b*c - a^2*d)*arctan(1/2*((b*c - 2*a*d)*x^6 - a*c)*sqrt(d*x^6 + c)*sqrt(a*b*c - a^2*
d)/((a*b*c*d - a^2*d^2)*x^9 + (a*b*c^2 - a^2*c*d)*x^3)) + 2*((3*a*b^3*c^2 - 5*a^2*b^2*c*d + 2*a^3*b*d^2)*x^6 +
2*a^2*b^2*c^2 - 4*a^3*b*c*d + 2*a^4*d^2)*sqrt(d*x^6 + c))/((a^3*b^3*c^3 - 2*a^4*b^2*c^2*d + a^5*b*c*d^2)*x^9
+ (a^4*b^2*c^3 - 2*a^5*b*c^2*d + a^6*c*d^2)*x^3)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}\, x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

[Out]

int(1/x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^6)^2*(c + d*x^6)^(1/2)),x)

[Out]

int(1/(x^4*(a + b*x^6)^2*(c + d*x^6)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)

[Out]

Timed out

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