3.8.26 \(\int \frac {x^7}{(a+b x^2)^{4/3}} \, dx\) [726]

Optimal. Leaf size=80 \[ \frac {3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac {9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac {9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac {3 \left (a+b x^2\right )^{8/3}}{16 b^4} \]

[Out]

3/2*a^3/b^4/(b*x^2+a)^(1/3)+9/4*a^2*(b*x^2+a)^(2/3)/b^4-9/10*a*(b*x^2+a)^(5/3)/b^4+3/16*(b*x^2+a)^(8/3)/b^4

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Rubi [A]
time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \frac {3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac {9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac {9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac {3 \left (a+b x^2\right )^{8/3}}{16 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^2)^(4/3),x]

[Out]

(3*a^3)/(2*b^4*(a + b*x^2)^(1/3)) + (9*a^2*(a + b*x^2)^(2/3))/(4*b^4) - (9*a*(a + b*x^2)^(5/3))/(10*b^4) + (3*
(a + b*x^2)^(8/3))/(16*b^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^7}{\left (a+b x^2\right )^{4/3}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{4/3}}+\frac {3 a^2}{b^3 \sqrt [3]{a+b x}}-\frac {3 a (a+b x)^{2/3}}{b^3}+\frac {(a+b x)^{5/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^3}{2 b^4 \sqrt [3]{a+b x^2}}+\frac {9 a^2 \left (a+b x^2\right )^{2/3}}{4 b^4}-\frac {9 a \left (a+b x^2\right )^{5/3}}{10 b^4}+\frac {3 \left (a+b x^2\right )^{8/3}}{16 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {3 \left (81 a^3+27 a^2 b x^2-9 a b^2 x^4+5 b^3 x^6\right )}{80 b^4 \sqrt [3]{a+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^2)^(4/3),x]

[Out]

(3*(81*a^3 + 27*a^2*b*x^2 - 9*a*b^2*x^4 + 5*b^3*x^6))/(80*b^4*(a + b*x^2)^(1/3))

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Maple [A]
time = 0.06, size = 47, normalized size = 0.59

method	result	size
gosper	\(\frac {\frac {3}{16} b^{3} x^{6}-\frac {27}{80} a \,b^{2} x^{4}+\frac {81}{80} a^{2} b \,x^{2}+\frac {243}{80} a^{3}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{4}}\)	\(47\)
trager	\(\frac {\frac {3}{16} b^{3} x^{6}-\frac {27}{80} a \,b^{2} x^{4}+\frac {81}{80} a^{2} b \,x^{2}+\frac {243}{80} a^{3}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{4}}\)	\(47\)
risch	\(\frac {3 \left (5 b^{2} x^{4}-14 a b \,x^{2}+41 a^{2}\right ) \left (b \,x^{2}+a \right )^{\frac {2}{3}}}{80 b^{4}}+\frac {3 a^{3}}{2 b^{4} \left (b \,x^{2}+a \right )^{\frac {1}{3}}}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^2+a)^(4/3),x,method=_RETURNVERBOSE)

[Out]

3/80/(b*x^2+a)^(1/3)*(5*b^3*x^6-9*a*b^2*x^4+27*a^2*b*x^2+81*a^3)/b^4

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Maxima [A]
time = 0.29, size = 64, normalized size = 0.80 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}}}{16 \, b^{4}} - \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a}{10 \, b^{4}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{2}}{4 \, b^{4}} + \frac {3 \, a^{3}}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

3/16*(b*x^2 + a)^(8/3)/b^4 - 9/10*(b*x^2 + a)^(5/3)*a/b^4 + 9/4*(b*x^2 + a)^(2/3)*a^2/b^4 + 3/2*a^3/((b*x^2 +
a)^(1/3)*b^4)

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Fricas [A]
time = 1.00, size = 58, normalized size = 0.72 \begin {gather*} \frac {3 \, {\left (5 \, b^{3} x^{6} - 9 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 81 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{80 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

3/80*(5*b^3*x^6 - 9*a*b^2*x^4 + 27*a^2*b*x^2 + 81*a^3)*(b*x^2 + a)^(2/3)/(b^5*x^2 + a*b^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1584 vs. \(2 (75) = 150\).
time = 1.38, size = 1584, normalized size = 19.80 \begin {gather*} \frac {243 a^{\frac {68}{3}} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {243 a^{\frac {68}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {1296 a^{\frac {65}{3}} b x^{2} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {1458 a^{\frac {65}{3}} b x^{2}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {2808 a^{\frac {62}{3}} b^{2} x^{4} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {3645 a^{\frac {62}{3}} b^{2} x^{4}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {3120 a^{\frac {59}{3}} b^{3} x^{6} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {4860 a^{\frac {59}{3}} b^{3} x^{6}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {1830 a^{\frac {56}{3}} b^{4} x^{8} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {3645 a^{\frac {56}{3}} b^{4} x^{8}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {528 a^{\frac {53}{3}} b^{5} x^{10} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {1458 a^{\frac {53}{3}} b^{5} x^{10}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {96 a^{\frac {50}{3}} b^{6} x^{12} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} - \frac {243 a^{\frac {50}{3}} b^{6} x^{12}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {48 a^{\frac {47}{3}} b^{7} x^{14} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} + \frac {15 a^{\frac {44}{3}} b^{8} x^{16} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {2}{3}}}{80 a^{20} b^{4} + 480 a^{19} b^{5} x^{2} + 1200 a^{18} b^{6} x^{4} + 1600 a^{17} b^{7} x^{6} + 1200 a^{16} b^{8} x^{8} + 480 a^{15} b^{9} x^{10} + 80 a^{14} b^{10} x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(b*x**2+a)**(4/3),x)

[Out]

243*a**(68/3)*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b
**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 243*a**(68/3)/(80*a**20*b**4
+ 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x*
*10 + 80*a**14*b**10*x**12) + 1296*a**(65/3)*b*x**2*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2
+ 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*
x**12) - 1458*a**(65/3)*b*x**2/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x
**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 2808*a**(62/3)*b**2*x**4*(1 + b*x*
*2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b
**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 3645*a**(62/3)*b**2*x**4/(80*a**20*b**4 + 480*a**19*
b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**
14*b**10*x**12) + 3120*a**(59/3)*b**3*x**6*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a
**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) -
4860*a**(59/3)*b**3*x**6/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 +
1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 1830*a**(56/3)*b**4*x**8*(1 + b*x**2/a)*
*(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x*
*8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 3645*a**(56/3)*b**4*x**8/(80*a**20*b**4 + 480*a**19*b**5*x
**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**
10*x**12) + 528*a**(53/3)*b**5*x**10*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b
**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 1458*a
**(53/3)*b**5*x**10/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*
a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 96*a**(50/3)*b**6*x**12*(1 + b*x**2/a)**(2/3)
/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 4
80*a**15*b**9*x**10 + 80*a**14*b**10*x**12) - 243*a**(50/3)*b**6*x**12/(80*a**20*b**4 + 480*a**19*b**5*x**2 +
1200*a**18*b**6*x**4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**
12) + 48*a**(47/3)*b**7*x**14*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**
4 + 1600*a**17*b**7*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12) + 15*a**(44/3)*
b**8*x**16*(1 + b*x**2/a)**(2/3)/(80*a**20*b**4 + 480*a**19*b**5*x**2 + 1200*a**18*b**6*x**4 + 1600*a**17*b**7
*x**6 + 1200*a**16*b**8*x**8 + 480*a**15*b**9*x**10 + 80*a**14*b**10*x**12)

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Giac [A]
time = 0.61, size = 70, normalized size = 0.88 \begin {gather*} \frac {3 \, a^{3}}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b^{4}} + \frac {3 \, {\left (5 \, {\left (b x^{2} + a\right )}^{\frac {8}{3}} b^{28} - 24 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} a b^{28} + 60 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{2} b^{28}\right )}}{80 \, b^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

3/2*a^3/((b*x^2 + a)^(1/3)*b^4) + 3/80*(5*(b*x^2 + a)^(8/3)*b^28 - 24*(b*x^2 + a)^(5/3)*a*b^28 + 60*(b*x^2 + a
)^(2/3)*a^2*b^28)/b^32

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Mupad [B]
time = 5.44, size = 55, normalized size = 0.69 \begin {gather*} \frac {180\,a^2\,\left (b\,x^2+a\right )-72\,a\,{\left (b\,x^2+a\right )}^2+15\,{\left (b\,x^2+a\right )}^3+120\,a^3}{80\,b^4\,{\left (b\,x^2+a\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^2)^(4/3),x)

[Out]

(180*a^2*(a + b*x^2) - 72*a*(a + b*x^2)^2 + 15*(a + b*x^2)^3 + 120*a^3)/(80*b^4*(a + b*x^2)^(1/3))

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