Integrand size = 22, antiderivative size = 340 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=-\frac {(b c-a d) x^3}{10 b^2 \left (a+b x^4\right )^{5/2}}+\frac {(3 b c-13 a d) x^3}{60 a b^2 \left (a+b x^4\right )^{3/2}}+\frac {(3 b c+7 a d) x^3}{40 a^2 b^2 \sqrt {a+b x^4}}-\frac {(3 b c+7 a d) x \sqrt {a+b x^4}}{40 a^2 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {(3 b c+7 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{40 a^{7/4} b^{11/4} \sqrt {a+b x^4}}-\frac {(3 b c+7 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{80 a^{7/4} b^{11/4} \sqrt {a+b x^4}} \] Output:
-1/10*(-a*d+b*c)*x^3/b^2/(b*x^4+a)^(5/2)+1/60*(-13*a*d+3*b*c)*x^3/a/b^2/(b *x^4+a)^(3/2)+1/40*(7*a*d+3*b*c)*x^3/a^2/b^2/(b*x^4+a)^(1/2)-1/40*(7*a*d+3 *b*c)*x*(b*x^4+a)^(1/2)/a^2/b^(5/2)/(a^(1/2)+b^(1/2)*x^2)+1/40*(7*a*d+3*b* c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*Ellipti cE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(7/4)/b^(11/4)/(b*x^4+a )^(1/2)-1/80*(7*a*d+3*b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/ 2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/ a^(7/4)/b^(11/4)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.29 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {x^3 \left (-a^2 \left (3 b c+7 a d+7 b d x^4\right )+(3 b c+7 a d) \left (a+b x^4\right )^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{21 a^2 b^2 \left (a+b x^4\right )^{5/2}} \] Input:
Integrate[(x^6*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
(x^3*(-(a^2*(3*b*c + 7*a*d + 7*b*d*x^4)) + (3*b*c + 7*a*d)*(a + b*x^4)^2*S qrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 7/2, 7/4, -((b*x^4)/a)]))/(21*a^ 2*b^2*(a + b*x^4)^(5/2))
Time = 0.67 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {957, 817, 819, 834, 27, 761, 1510}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(7 a d+3 b c) \int \frac {x^6}{\left (b x^4+a\right )^{5/2}}dx}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 817 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\int \frac {x^2}{\left (b x^4+a\right )^{3/2}}dx}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 819 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\int \frac {x^2}{\sqrt {b x^4+a}}dx}{2 a}}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 834 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {a} \sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt {a} \int \frac {1}{\sqrt {b x^4+a}}dx}{\sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{\sqrt {b x^4+a}}dx}{\sqrt {b}}}{2 a}}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
\(\Big \downarrow \) 1510 |
\(\displaystyle \frac {(7 a d+3 b c) \left (\frac {\frac {x^3}{2 a \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^4}}-\frac {x \sqrt {a+b x^4}}{\sqrt {a}+\sqrt {b} x^2}}{\sqrt {b}}}{2 a}}{2 b}-\frac {x^3}{6 b \left (a+b x^4\right )^{3/2}}\right )}{10 a b}+\frac {x^7 (b c-a d)}{10 a b \left (a+b x^4\right )^{5/2}}\) |
Input:
Int[(x^6*(c + d*x^4))/(a + b*x^4)^(7/2),x]
Output:
((b*c - a*d)*x^7)/(10*a*b*(a + b*x^4)^(5/2)) + ((3*b*c + 7*a*d)*(-1/6*x^3/ (b*(a + b*x^4)^(3/2)) + (x^3/(2*a*Sqrt[a + b*x^4]) - (-((-((x*Sqrt[a + b*x ^4])/(Sqrt[a] + Sqrt[b]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)] , 1/2])/(b^(1/4)*Sqrt[a + b*x^4]))/Sqrt[b]) + (a^(1/4)*(Sqrt[a] + Sqrt[b]* x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/ 4)*x)/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^4]))/(2*a))/(2*b)))/(10*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.64
method | result | size |
elliptic | \(\frac {x^{3} \left (a d -c b \right ) \sqrt {b \,x^{4}+a}}{10 b^{5} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {x^{3} \left (13 a d -3 c b \right ) \sqrt {b \,x^{4}+a}}{60 a \,b^{4} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {x^{3} \left (7 a d +3 c b \right )}{40 b^{2} a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \left (7 a d +3 c b \right ) \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 b^{\frac {5}{2}} a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(216\) |
default | \(c \left (-\frac {x^{3} \sqrt {b \,x^{4}+a}}{10 b^{4} \left (x^{4}+\frac {a}{b}\right )^{3}}+\frac {x^{3} \sqrt {b \,x^{4}+a}}{20 a \,b^{3} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {3 x^{3}}{40 b \,a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {3 i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 b^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {a \,x^{3} \sqrt {b \,x^{4}+a}}{10 b^{5} \left (x^{4}+\frac {a}{b}\right )^{3}}-\frac {13 x^{3} \sqrt {b \,x^{4}+a}}{60 b^{4} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {7 x^{3}}{40 b^{2} a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {7 i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{40 b^{\frac {5}{2}} \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(364\) |
Input:
int(x^6*(d*x^4+c)/(b*x^4+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/10*x^3/b^5*(a*d-b*c)*(b*x^4+a)^(1/2)/(x^4+a/b)^3-1/60/a*x^3*(13*a*d-3*b* c)/b^4*(b*x^4+a)^(1/2)/(x^4+a/b)^2+1/40/b^2/a^2*x^3*(7*a*d+3*b*c)/((x^4+a/ b)*b)^(1/2)-1/40*I/b^(5/2)/a^(3/2)*(7*a*d+3*b*c)/(I/a^(1/2)*b^(1/2))^(1/2) *(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a) ^(1/2)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^ (1/2))^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.95 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {3 \, {\left ({\left (3 \, b^{4} c + 7 \, a b^{3} d\right )} x^{12} + 3 \, {\left (3 \, a b^{3} c + 7 \, a^{2} b^{2} d\right )} x^{8} + 3 \, a^{3} b c + 7 \, a^{4} d + 3 \, {\left (3 \, a^{2} b^{2} c + 7 \, a^{3} b d\right )} x^{4}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3 \, {\left ({\left (3 \, b^{4} c + 7 \, a b^{3} d\right )} x^{12} + 3 \, {\left (3 \, a b^{3} c + 7 \, a^{2} b^{2} d\right )} x^{8} + 3 \, a^{3} b c + 7 \, a^{4} d + 3 \, {\left (3 \, a^{2} b^{2} c + 7 \, a^{3} b d\right )} x^{4}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, {\left (3 \, b^{4} c + 7 \, a b^{3} d\right )} x^{11} + 8 \, {\left (3 \, a b^{3} c + 2 \, a^{2} b^{2} d\right )} x^{7} + {\left (3 \, a^{2} b^{2} c + 7 \, a^{3} b d\right )} x^{3}\right )} \sqrt {b x^{4} + a}}{120 \, {\left (a^{2} b^{6} x^{12} + 3 \, a^{3} b^{5} x^{8} + 3 \, a^{4} b^{4} x^{4} + a^{5} b^{3}\right )}} \] Input:
integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="fricas")
Output:
1/120*(3*((3*b^4*c + 7*a*b^3*d)*x^12 + 3*(3*a*b^3*c + 7*a^2*b^2*d)*x^8 + 3 *a^3*b*c + 7*a^4*d + 3*(3*a^2*b^2*c + 7*a^3*b*d)*x^4)*sqrt(a)*(-b/a)^(3/4) *elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 3*((3*b^4*c + 7*a*b^3*d)*x^12 + 3*(3*a*b^3*c + 7*a^2*b^2*d)*x^8 + 3*a^3*b*c + 7*a^4*d + 3*(3*a^2*b^2*c + 7 *a^3*b*d)*x^4)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + (3*(3*b^4*c + 7*a*b^3*d)*x^11 + 8*(3*a*b^3*c + 2*a^2*b^2*d)*x^7 + (3*a^ 2*b^2*c + 7*a^3*b*d)*x^3)*sqrt(b*x^4 + a))/(a^2*b^6*x^12 + 3*a^3*b^5*x^8 + 3*a^4*b^4*x^4 + a^5*b^3)
Result contains complex when optimal does not.
Time = 114.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.24 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {c x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {d x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {11}{4}, \frac {7}{2} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{2}} \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate(x**6*(d*x**4+c)/(b*x**4+a)**(7/2),x)
Output:
c*x**7*gamma(7/4)*hyper((7/4, 7/2), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4* a**(7/2)*gamma(11/4)) + d*x**11*gamma(11/4)*hyper((11/4, 7/2), (15/4,), b* x**4*exp_polar(I*pi)/a)/(4*a**(7/2)*gamma(15/4))
\[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{6}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)*x^6/(b*x^4 + a)^(7/2), x)
\[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{6}}{{\left (b x^{4} + a\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/2),x, algorithm="giac")
Output:
integrate((d*x^4 + c)*x^6/(b*x^4 + a)^(7/2), x)
Timed out. \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\int \frac {x^6\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/2}} \,d x \] Input:
int((x^6*(c + d*x^4))/(a + b*x^4)^(7/2),x)
Output:
int((x^6*(c + d*x^4))/(a + b*x^4)^(7/2), x)
\[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/2}} \, dx=\frac {-7 \sqrt {b \,x^{4}+a}\, a d \,x^{3}-3 \sqrt {b \,x^{4}+a}\, b c \,x^{3}-7 \sqrt {b \,x^{4}+a}\, b d \,x^{7}+21 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{5} d +9 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b c +63 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{4} b d \,x^{4}+27 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{3} b^{2} c \,x^{4}+63 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{3} b^{2} d \,x^{8}+27 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{2} b^{3} c \,x^{8}+21 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a^{2} b^{3} d \,x^{12}+9 \left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{4} x^{16}+4 a \,b^{3} x^{12}+6 a^{2} b^{2} x^{8}+4 a^{3} b \,x^{4}+a^{4}}d x \right ) a \,b^{4} c \,x^{12}}{21 b^{2} \left (b^{3} x^{12}+3 a \,b^{2} x^{8}+3 a^{2} b \,x^{4}+a^{3}\right )} \] Input:
int(x^6*(d*x^4+c)/(b*x^4+a)^(7/2),x)
Output:
( - 7*sqrt(a + b*x**4)*a*d*x**3 - 3*sqrt(a + b*x**4)*b*c*x**3 - 7*sqrt(a + b*x**4)*b*d*x**7 + 21*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**5*d + 9*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**4*b*c + 63*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3 *b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**4*b*d*x**4 + 27*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**3*b**2*c*x**4 + 63*int((sqrt(a + b*x **4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b** 4*x**16),x)*a**3*b**2*d*x**8 + 27*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a* *3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**2*b**3*c *x**8 + 21*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2 *x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a**2*b**3*d*x**12 + 9*int((sqrt(a + b*x**4)*x**2)/(a**4 + 4*a**3*b*x**4 + 6*a**2*b**2*x**8 + 4*a*b**3*x**12 + b**4*x**16),x)*a*b**4*c*x**12)/(21*b**2*(a**3 + 3*a**2*b*x**4 + 3*a*b**2 *x**8 + b**3*x**12))