3.814 \(\int \frac{1}{x^7 (a+b x^4) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=115 \[ \frac{b^2 \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac{\sqrt{c+d x^4}}{6 a c x^6} \]

[Out]

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

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Rubi [A]  time = 0.163556, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {465, 480, 583, 12, 377, 205} \[ \frac{b^2 \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4} (2 a d+3 b c)}{6 a^2 c^2 x^2}-\frac{\sqrt{c+d x^4}}{6 a c x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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 480

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

Rule 12

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

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 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]

Rubi steps

\begin{align*} \int \frac{1}{x^7 \left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c+d x^4}}{6 a c x^6}+\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^2\right )}{6 a c}\\ &=-\frac{\sqrt{c+d x^4}}{6 a c x^6}+\frac{(3 b c+2 a d) \sqrt{c+d x^4}}{6 a^2 c^2 x^2}-\frac{\operatorname{Subst}\left (\int -\frac{3 b^2 c^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{6 a^2 c^2}\\ &=-\frac{\sqrt{c+d x^4}}{6 a c x^6}+\frac{(3 b c+2 a d) \sqrt{c+d x^4}}{6 a^2 c^2 x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac{\sqrt{c+d x^4}}{6 a c x^6}+\frac{(3 b c+2 a d) \sqrt{c+d x^4}}{6 a^2 c^2 x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x^2}{\sqrt{c+d x^4}}\right )}{2 a^2}\\ &=-\frac{\sqrt{c+d x^4}}{6 a c x^6}+\frac{(3 b c+2 a d) \sqrt{c+d x^4}}{6 a^2 c^2 x^2}+\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{b c-a d} x^2}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 a^{5/2} \sqrt{b c-a d}}\\ \end{align*}

Mathematica [C]  time = 1.58436, size = 253, normalized size = 2.2 \[ -\frac{\left (\frac{d x^4}{c}+1\right ) \left (-\frac{8 x^4 \left (c+d x^4\right )^2 (b c-a d) \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{5}{2}\right \},\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}\right )}{a+b x^4}+\frac{3 c \left (c^2-4 c d x^4-8 d^2 x^8\right ) \sin ^{-1}\left (\sqrt{\frac{x^4 (b c-a d)}{c \left (a+b x^4\right )}}\right )}{\sqrt{\frac{a x^4 \left (c+d x^4\right ) (b c-a d)}{c^2 \left (a+b x^4\right )^2}}}+\frac{24 d x^8 \left (c+d x^4\right ) (a d-b c) \, _2F_1\left (2,2;\frac{5}{2};\frac{(b c-a d) x^4}{c \left (b x^4+a\right )}\right )}{a+b x^4}\right )}{18 c^3 x^6 \left (a+b x^4\right ) \sqrt{c+d x^4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x^7*(a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

-((1 + (d*x^4)/c)*((3*c*(c^2 - 4*c*d*x^4 - 8*d^2*x^8)*ArcSin[Sqrt[((b*c - a*d)*x^4)/(c*(a + b*x^4))]])/Sqrt[(a
*(b*c - a*d)*x^4*(c + d*x^4))/(c^2*(a + b*x^4)^2)] + (24*d*(-(b*c) + a*d)*x^8*(c + d*x^4)*Hypergeometric2F1[2,
2, 5/2, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(a + b*x^4) - (8*(b*c - a*d)*x^4*(c + d*x^4)^2*HypergeometricPFQ[
{2, 2, 2}, {1, 5/2}, ((b*c - a*d)*x^4)/(c*(a + b*x^4))])/(a + b*x^4)))/(18*c^3*x^6*(a + b*x^4)*Sqrt[c + d*x^4]
)

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Maple [B]  time = 0.015, size = 383, normalized size = 3.3 \begin{align*}{\frac{b}{2\,{a}^{2}{x}^{2}c}\sqrt{d{x}^{4}+c}}-{\frac{{b}^{2}}{4\,{a}^{2}}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{{b}^{2}}{4\,{a}^{2}}\ln \left ({ \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{-2\,d{x}^{4}+c}{6\,a{x}^{6}{c}^{2}}\sqrt{d{x}^{4}+c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

1/2/a^2*b*(d*x^4+c)^(1/2)/x^2/c-1/4*b^2/a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1
/2)/b*(x^2-(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^2-(-a*b)^(1/2)/b)^2*d+2*d*(-a*b)^(1/2)/b*(x^2-(-a*b)^(1/
2)/b)-(a*d-b*c)/b)^(1/2))/(x^2-(-a*b)^(1/2)/b))+1/4*b^2/a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)
/b-2*d*(-a*b)^(1/2)/b*(x^2+(-a*b)^(1/2)/b)+2*(-(a*d-b*c)/b)^(1/2)*((x^2+(-a*b)^(1/2)/b)^2*d-2*d*(-a*b)^(1/2)/b
*(x^2+(-a*b)^(1/2)/b)-(a*d-b*c)/b)^(1/2))/(x^2+(-a*b)^(1/2)/b))-1/6/a*(d*x^4+c)^(1/2)*(-2*d*x^4+c)/x^6/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(d*x^4 + c)*x^7), x)

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Fricas [A]  time = 2.3072, size = 856, normalized size = 7.44 \begin{align*} \left [-\frac{3 \, \sqrt{-a b c + a^{2} d} b^{2} c^{2} x^{6} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (b c - 2 \, a d\right )} x^{6} - a c x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \,{\left (a^{2} b c^{2} - a^{3} c d -{\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{24 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}, \frac{3 \, \sqrt{a b c - a^{2} d} b^{2} c^{2} x^{6} \arctan \left (\frac{{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d}}{2 \,{\left ({\left (a b c d - a^{2} d^{2}\right )} x^{6} +{\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )}}\right ) - 2 \,{\left (a^{2} b c^{2} - a^{3} c d -{\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{4}\right )} \sqrt{d x^{4} + c}}{12 \,{\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{7} \left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(1/(x**7*(a + b*x**4)*sqrt(c + d*x**4)), x)

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Giac [A]  time = 1.10981, size = 144, normalized size = 1.25 \begin{align*} -\frac{b^{2} \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{2 \, \sqrt{a b c - a^{2} d} a^{2}} + \frac{3 \, a b c^{5} \sqrt{d + \frac{c}{x^{4}}} - a^{2} c^{4}{\left (d + \frac{c}{x^{4}}\right )}^{\frac{3}{2}} + 3 \, a^{2} c^{4} \sqrt{d + \frac{c}{x^{4}}} d}{6 \, a^{3} c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

-1/2*b^2*arctan(a*sqrt(d + c/x^4)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*a^2) + 1/6*(3*a*b*c^5*sqrt(d + c/x
^4) - a^2*c^4*(d + c/x^4)^(3/2) + 3*a^2*c^4*sqrt(d + c/x^4)*d)/(a^3*c^6)