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

Optimal. Leaf size=93 \[ \frac {a \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}}-\frac {a \sqrt {c+d x^4}}{2 b^2}+\frac {\left (c+d x^4\right )^{3/2}}{6 b d} \]

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Rubi [A]  time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 80, 50, 63, 208} \begin {gather*} -\frac {a \sqrt {c+d x^4}}{2 b^2}+\frac {a \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}}+\frac {\left (c+d x^4\right )^{3/2}}{6 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 50

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

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

Rule 208

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

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]]

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 88, normalized size = 0.95 \begin {gather*} \frac {a \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}}+\frac {\sqrt {c+d x^4} \left (b \left (c+d x^4\right )-3 a d\right )}{6 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.11, size = 98, normalized size = 1.05 \begin {gather*} \frac {\sqrt {c+d x^4} \left (-3 a d+b c+b d x^4\right )}{6 b^2 d}-\frac {a \sqrt {a d-b c} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4} \sqrt {a d-b c}}{b c-a d}\right )}{2 b^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 195, normalized size = 2.10 \begin {gather*} \left [\frac {3 \, a d \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x^{4} + 2 \, b c - a d + 2 \, \sqrt {d x^{4} + c} b \sqrt {\frac {b c - a d}{b}}}{b x^{4} + a}\right ) + 2 \, {\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt {d x^{4} + c}}{12 \, b^{2} d}, \frac {3 \, a d \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x^{4} + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) + {\left (b d x^{4} + b c - 3 \, a d\right )} \sqrt {d x^{4} + c}}{6 \, b^{2} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 96, normalized size = 1.03 \begin {gather*} -\frac {{\left (a b c - a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{4} + c\right )}^{\frac {3}{2}} b^{2} d^{2} - 3 \, \sqrt {d x^{4} + c} a b d^{3}}{6 \, b^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.27, size = 1015, normalized size = 10.91 \begin {gather*} -\frac {a^{2} d \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{3}}-\frac {a^{2} d \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{3}}+\frac {a c \ln \left (\frac {\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{2}}+\frac {a c \ln \left (\frac {-\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {2 \left (a d -b c \right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{4 \sqrt {-\frac {a d -b c}{b}}\, b^{2}}+\frac {\sqrt {-a b}\, a \sqrt {d}\, \ln \left (\frac {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d -\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\right )}{4 b^{3}}-\frac {\sqrt {-a b}\, a \sqrt {d}\, \ln \left (\frac {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d +\frac {\sqrt {-a b}\, d}{b}}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\right )}{4 b^{3}}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}{4 b^{2}}-\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{b}-\frac {a d -b c}{b}}\, a}{4 b^{2}}+\frac {\left (d \,x^{4}+c \right )^{\frac {3}{2}}}{6 b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(d*x^4+c)^(1/2)/(b*x^4+a),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 details)Is a*d-b*c positive or negative?

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mupad [B]  time = 4.69, size = 87, normalized size = 0.94 \begin {gather*} \frac {{\left (d\,x^4+c\right )}^{3/2}}{6\,b\,d}-\frac {a\,\sqrt {d\,x^4+c}}{2\,b^2}+\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^4+c}\,\sqrt {a\,d-b\,c}}{a^2\,d-a\,b\,c}\right )\,\sqrt {a\,d-b\,c}}{2\,b^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 17.44, size = 90, normalized size = 0.97 \begin {gather*} \frac {2 \left (- \frac {a d^{2} \sqrt {c + d x^{4}}}{4 b^{2}} + \frac {a d^{2} \left (a d - b c\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x^{4}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{4 b^{3} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x^{4}\right )^{\frac {3}{2}}}{12 b}\right )}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-a*d**2*sqrt(c + d*x**4)/(4*b**2) + a*d**2*(a*d - b*c)*atan(sqrt(c + d*x**4)/sqrt((a*d - b*c)/b))/(4*b**3*s
qrt((a*d - b*c)/b)) + d*(c + d*x**4)**(3/2)/(12*b))/d**2

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