3.219 \(\int \frac {x^4 (c+d x^2)^3}{a+b x^2} \, dx\)

Optimal. Leaf size=140 \[ \frac {a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}-\frac {a x (b c-a d)^3}{b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^7 (3 b c-a d)}{7 b^2}+\frac {d^3 x^9}{9 b} \]

[Out]

-a*(-a*d+b*c)^3*x/b^5+1/3*(-a*d+b*c)^3*x^3/b^4+1/5*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^5/b^3+1/7*d^2*(-a*d+3*b*c
)*x^7/b^2+1/9*d^3*x^9/b+a^(3/2)*(-a*d+b*c)^3*arctan(x*b^(1/2)/a^(1/2))/b^(11/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {461, 205} \[ \frac {d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac {a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {d^2 x^7 (3 b c-a d)}{7 b^2}+\frac {x^3 (b c-a d)^3}{3 b^4}-\frac {a x (b c-a d)^3}{b^5}+\frac {d^3 x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

-((a*(b*c - a*d)^3*x)/b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) +
(d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b) + (a^(3/2)*(b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11
/2)

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 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rubi steps

\begin {align*} \int \frac {x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (-\frac {a (b c-a d)^3}{b^5}+\frac {(b c-a d)^3 x^2}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{b^3}+\frac {d^2 (3 b c-a d) x^6}{b^2}+\frac {d^3 x^8}{b}+\frac {a^2 b^3 c^3-3 a^3 b^2 c^2 d+3 a^4 b c d^2-a^5 d^3}{b^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^2 (3 b c-a d) x^7}{7 b^2}+\frac {d^3 x^9}{9 b}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{a+b x^2} \, dx}{b^5}\\ &=-\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac {d^2 (3 b c-a d) x^7}{7 b^2}+\frac {d^3 x^9}{9 b}+\frac {a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 140, normalized size = 1.00 \[ -\frac {a^{3/2} (a d-b c)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}+\frac {d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac {a x (a d-b c)^3}{b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^7 (3 b c-a d)}{7 b^2}+\frac {d^3 x^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(a*(-(b*c) + a*d)^3*x)/b^5 + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) +
(d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b) - (a^(3/2)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^
(11/2)

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 468, normalized size = 3.34 \[ \left [\frac {70 \, b^{4} d^{3} x^{9} + 90 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 126 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 210 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{630 \, b^{5}}, \frac {35 \, b^{4} d^{3} x^{9} + 45 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/630*(70*b^4*d^3*x^9 + 90*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 126*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^
5 + 210*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3 - 315*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3
*b*c*d^2 - a^4*d^3)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 630*(a*b^3*c^3 - 3*a^2*b^2*c^
2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5, 1/315*(35*b^4*d^3*x^9 + 45*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 63*(3*b^4*c^
2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 105*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3 + 315
*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(a*b^3*c^3 -
3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5]

________________________________________________________________________________________

giac [A]  time = 0.30, size = 241, normalized size = 1.72 \[ \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{8} d^{3} x^{9} + 135 \, b^{8} c d^{2} x^{7} - 45 \, a b^{7} d^{3} x^{7} + 189 \, b^{8} c^{2} d x^{5} - 189 \, a b^{7} c d^{2} x^{5} + 63 \, a^{2} b^{6} d^{3} x^{5} + 105 \, b^{8} c^{3} x^{3} - 315 \, a b^{7} c^{2} d x^{3} + 315 \, a^{2} b^{6} c d^{2} x^{3} - 105 \, a^{3} b^{5} d^{3} x^{3} - 315 \, a b^{7} c^{3} x + 945 \, a^{2} b^{6} c^{2} d x - 945 \, a^{3} b^{5} c d^{2} x + 315 \, a^{4} b^{4} d^{3} x}{315 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^
8*d^3*x^9 + 135*b^8*c*d^2*x^7 - 45*a*b^7*d^3*x^7 + 189*b^8*c^2*d*x^5 - 189*a*b^7*c*d^2*x^5 + 63*a^2*b^6*d^3*x^
5 + 105*b^8*c^3*x^3 - 315*a*b^7*c^2*d*x^3 + 315*a^2*b^6*c*d^2*x^3 - 105*a^3*b^5*d^3*x^3 - 315*a*b^7*c^3*x + 94
5*a^2*b^6*c^2*d*x - 945*a^3*b^5*c*d^2*x + 315*a^4*b^4*d^3*x)/b^9

________________________________________________________________________________________

maple [B]  time = 0.01, size = 276, normalized size = 1.97 \[ \frac {d^{3} x^{9}}{9 b}-\frac {a \,d^{3} x^{7}}{7 b^{2}}+\frac {3 c \,d^{2} x^{7}}{7 b}+\frac {a^{2} d^{3} x^{5}}{5 b^{3}}-\frac {3 a c \,d^{2} x^{5}}{5 b^{2}}+\frac {3 c^{2} d \,x^{5}}{5 b}-\frac {a^{3} d^{3} x^{3}}{3 b^{4}}+\frac {a^{2} c \,d^{2} x^{3}}{b^{3}}-\frac {a \,c^{2} d \,x^{3}}{b^{2}}+\frac {c^{3} x^{3}}{3 b}-\frac {a^{5} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {3 a^{4} c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {3 a^{3} c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {a^{4} d^{3} x}{b^{5}}-\frac {3 a^{3} c \,d^{2} x}{b^{4}}+\frac {3 a^{2} c^{2} d x}{b^{3}}-\frac {a \,c^{3} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

1/9*d^3*x^9/b-1/7/b^2*x^7*a*d^3+3/7/b*x^7*c*d^2+1/5/b^3*x^5*a^2*d^3-3/5/b^2*x^5*a*c*d^2+3/5/b*x^5*c^2*d-1/3/b^
4*x^3*a^3*d^3+1/b^3*x^3*a^2*c*d^2-1/b^2*x^3*a*c^2*d+1/3/b*x^3*c^3+1/b^5*a^4*d^3*x-3/b^4*a^3*c*d^2*x+3/b^3*a^2*
c^2*d*x-1/b^2*a*c^3*x-a^5/b^5/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*d^3+3*a^4/b^4/(a*b)^(1/2)*arctan(1/(a*b)^(
1/2)*b*x)*c*d^2-3*a^3/b^3/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c^2*d+a^2/b^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)
*b*x)*c^3

________________________________________________________________________________________

maxima [A]  time = 2.58, size = 222, normalized size = 1.59 \[ \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{4} d^{3} x^{9} + 45 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^
4*d^3*x^9 + 45*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 63*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 105*(b^4*c
^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3 - 315*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4
*d^3)*x)/b^5

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 260, normalized size = 1.86 \[ x^3\,\left (\frac {c^3}{3\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^7\,\left (\frac {a\,d^3}{7\,b^2}-\frac {3\,c\,d^2}{7\,b}\right )+x^5\,\left (\frac {3\,c^2\,d}{5\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{5\,b}\right )+\frac {d^3\,x^9}{9\,b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^3}{b^{11/2}}-\frac {a\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(c + d*x^2)^3)/(a + b*x^2),x)

[Out]

x^3*(c^3/(3*b) - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/(3*b)) - x^7*((a*d^3)/(7*b^2) - (3*c*d^
2)/(7*b)) + x^5*((3*c^2*d)/(5*b) + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/(5*b)) + (d^3*x^9)/(9*b) - (a^(3/2)*atan((a
^(3/2)*b^(1/2)*x*(a*d - b*c)^3)/(a^5*d^3 - a^2*b^3*c^3 + 3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2))*(a*d - b*c)^3)/b^(1
1/2) - (a*x*(c^3/b - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b))/b

________________________________________________________________________________________

sympy [B]  time = 0.72, size = 343, normalized size = 2.45 \[ x^{7} \left (- \frac {a d^{3}}{7 b^{2}} + \frac {3 c d^{2}}{7 b}\right ) + x^{5} \left (\frac {a^{2} d^{3}}{5 b^{3}} - \frac {3 a c d^{2}}{5 b^{2}} + \frac {3 c^{2} d}{5 b}\right ) + x^{3} \left (- \frac {a^{3} d^{3}}{3 b^{4}} + \frac {a^{2} c d^{2}}{b^{3}} - \frac {a c^{2} d}{b^{2}} + \frac {c^{3}}{3 b}\right ) + x \left (\frac {a^{4} d^{3}}{b^{5}} - \frac {3 a^{3} c d^{2}}{b^{4}} + \frac {3 a^{2} c^{2} d}{b^{3}} - \frac {a c^{3}}{b^{2}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log {\left (- \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log {\left (\frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{9}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

x**7*(-a*d**3/(7*b**2) + 3*c*d**2/(7*b)) + x**5*(a**2*d**3/(5*b**3) - 3*a*c*d**2/(5*b**2) + 3*c**2*d/(5*b)) +
x**3*(-a**3*d**3/(3*b**4) + a**2*c*d**2/b**3 - a*c**2*d/b**2 + c**3/(3*b)) + x*(a**4*d**3/b**5 - 3*a**3*c*d**2
/b**4 + 3*a**2*c**2*d/b**3 - a*c**3/b**2) + sqrt(-a**3/b**11)*(a*d - b*c)**3*log(-b**5*sqrt(-a**3/b**11)*(a*d
- b*c)**3/(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3) + x)/2 - sqrt(-a**3/b**11)*(a*d - b
*c)**3*log(b**5*sqrt(-a**3/b**11)*(a*d - b*c)**3/(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c*
*3) + x)/2 + d**3*x**9/(9*b)

________________________________________________________________________________________