3.2.94 \(\int \frac {x^4}{(b x^{2/3}+a x)^{3/2}} \, dx\) [194]

Optimal. Leaf size=336 \[ -\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {524288 b^9 \sqrt {b x^{2/3}+a x}}{29393 a^{11}}+\frac {1048576 b^{10} \sqrt {b x^{2/3}+a x}}{29393 a^{12} \sqrt [3]{x}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2} \]

[Out]

-6*x^4/a/(b*x^(2/3)+a*x)^(1/2)-524288/29393*b^9*(b*x^(2/3)+a*x)^(1/2)/a^11+1048576/29393*b^10*(b*x^(2/3)+a*x)^
(1/2)/a^12/x^(1/3)+393216/29393*b^8*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^10-327680/29393*b^7*x^(2/3)*(b*x^(2/3)+a*x
)^(1/2)/a^9+40960/4199*b^6*x*(b*x^(2/3)+a*x)^(1/2)/a^8-36864/4199*b^5*x^(4/3)*(b*x^(2/3)+a*x)^(1/2)/a^7+33792/
4199*b^4*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a^6-16896/2261*b^3*x^2*(b*x^(2/3)+a*x)^(1/2)/a^5+15840/2261*b^2*x^(7/3)
*(b*x^(2/3)+a*x)^(1/2)/a^4-880/133*b*x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a^3+44/7*x^3*(b*x^(2/3)+a*x)^(1/2)/a^2

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Rubi [A]
time = 0.39, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2040, 2041, 2027, 2039} \begin {gather*} \frac {1048576 b^{10} \sqrt {a x+b x^{2/3}}}{29393 a^{12} \sqrt [3]{x}}-\frac {524288 b^9 \sqrt {a x+b x^{2/3}}}{29393 a^{11}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {a x+b x^{2/3}}}{29393 a^9}+\frac {40960 b^6 x \sqrt {a x+b x^{2/3}}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {a x+b x^{2/3}}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {a x+b x^{2/3}}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {a x+b x^{2/3}}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {a x+b x^{2/3}}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {a x+b x^{2/3}}}{133 a^3}+\frac {44 x^3 \sqrt {a x+b x^{2/3}}}{7 a^2}-\frac {6 x^4}{a \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) - (524288*b^9*Sqrt[b*x^(2/3) + a*x])/(29393*a^11) + (1048576*b^10*Sqrt[b*x^
(2/3) + a*x])/(29393*a^12*x^(1/3)) + (393216*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^10) - (327680*b^7*x^(
2/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^9) + (40960*b^6*x*Sqrt[b*x^(2/3) + a*x])/(4199*a^8) - (36864*b^5*x^(4/3)*
Sqrt[b*x^(2/3) + a*x])/(4199*a^7) + (33792*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^6) - (16896*b^3*x^2*Sqrt
[b*x^(2/3) + a*x])/(2261*a^5) + (15840*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) - (880*b*x^(8/3)*Sqrt[b*x
^(2/3) + a*x])/(133*a^3) + (44*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a^2)

Rule 2027

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j -
1)), x] - Dist[b*((n*p + n - j + 1)/(a*(j*p + 1))), Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j,
n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n - j)], 0] && NeQ[j*p + 1, 0]

Rule 2039

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2040

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-c^(j - 1))*(c*x)^(m - j
+ 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j)*(p + 1))), x] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n,
j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])

Rule 2041

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(j - 1)*(c*x)^(m - j +
1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rubi steps

\begin {align*} \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {22 \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}-\frac {(440 b) \int \frac {x^{8/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a^2}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}+\frac {\left (2640 b^2\right ) \int \frac {x^{7/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{133 a^3}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}-\frac {\left (42240 b^3\right ) \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{2261 a^4}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}+\frac {\left (5632 b^4\right ) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{323 a^5}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}-\frac {\left (67584 b^5\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{4199 a^6}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}+\frac {\left (61440 b^6\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{4199 a^7}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}-\frac {\left (163840 b^7\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{12597 a^8}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}+\frac {\left (327680 b^8\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{29393 a^9}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}-\frac {\left (262144 b^9\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{29393 a^{10}}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {524288 b^9 \sqrt {b x^{2/3}+a x}}{29393 a^{11}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}+\frac {\left (524288 b^{10}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{88179 a^{11}}\\ &=-\frac {6 x^4}{a \sqrt {b x^{2/3}+a x}}-\frac {524288 b^9 \sqrt {b x^{2/3}+a x}}{29393 a^{11}}+\frac {1048576 b^{10} \sqrt {b x^{2/3}+a x}}{29393 a^{12} \sqrt [3]{x}}+\frac {393216 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{29393 a^{10}}-\frac {327680 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{29393 a^9}+\frac {40960 b^6 x \sqrt {b x^{2/3}+a x}}{4199 a^8}-\frac {36864 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{4199 a^7}+\frac {33792 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^6}-\frac {16896 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^5}+\frac {15840 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^4}-\frac {880 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^3}+\frac {44 x^3 \sqrt {b x^{2/3}+a x}}{7 a^2}\\ \end {align*}

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Mathematica [A]
time = 4.18, size = 161, normalized size = 0.48 \begin {gather*} \frac {2 \sqrt [3]{x} \left (524288 b^{11}+262144 a b^{10} \sqrt [3]{x}-65536 a^2 b^9 x^{2/3}+32768 a^3 b^8 x-20480 a^4 b^7 x^{4/3}+14336 a^5 b^6 x^{5/3}-10752 a^6 b^5 x^2+8448 a^7 b^4 x^{7/3}-6864 a^8 b^3 x^{8/3}+5720 a^9 b^2 x^3-4862 a^{10} b x^{10/3}+4199 a^{11} x^{11/3}\right )}{29393 a^{12} \sqrt {b x^{2/3}+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(1/3)*(524288*b^11 + 262144*a*b^10*x^(1/3) - 65536*a^2*b^9*x^(2/3) + 32768*a^3*b^8*x - 20480*a^4*b^7*x^(4
/3) + 14336*a^5*b^6*x^(5/3) - 10752*a^6*b^5*x^2 + 8448*a^7*b^4*x^(7/3) - 6864*a^8*b^3*x^(8/3) + 5720*a^9*b^2*x
^3 - 4862*a^10*b*x^(10/3) + 4199*a^11*x^(11/3)))/(29393*a^12*Sqrt[b*x^(2/3) + a*x])

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Maple [A]
time = 0.39, size = 143, normalized size = 0.43

method	result	size
derivativedivides	\(\frac {2 x \left (b +a \,x^{\frac {1}{3}}\right ) \left (4199 a^{11} x^{\frac {11}{3}}-4862 a^{10} b \,x^{\frac {10}{3}}+5720 a^{9} b^{2} x^{3}-6864 a^{8} b^{3} x^{\frac {8}{3}}+8448 a^{7} b^{4} x^{\frac {7}{3}}-10752 a^{6} b^{5} x^{2}+14336 a^{5} b^{6} x^{\frac {5}{3}}-20480 a^{4} b^{7} x^{\frac {4}{3}}+32768 a^{3} b^{8} x -65536 a^{2} b^{9} x^{\frac {2}{3}}+262144 a \,b^{10} x^{\frac {1}{3}}+524288 b^{11}\right )}{29393 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{12}}\)	\(143\)
default	\(\frac {2 x \left (b +a \,x^{\frac {1}{3}}\right ) \left (4199 a^{11} x^{\frac {11}{3}}-4862 a^{10} b \,x^{\frac {10}{3}}+5720 a^{9} b^{2} x^{3}-6864 a^{8} b^{3} x^{\frac {8}{3}}+8448 a^{7} b^{4} x^{\frac {7}{3}}-10752 a^{6} b^{5} x^{2}+14336 a^{5} b^{6} x^{\frac {5}{3}}-20480 a^{4} b^{7} x^{\frac {4}{3}}+32768 a^{3} b^{8} x -65536 a^{2} b^{9} x^{\frac {2}{3}}+262144 a \,b^{10} x^{\frac {1}{3}}+524288 b^{11}\right )}{29393 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} a^{12}}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/29393*x*(b+a*x^(1/3))*(4199*a^11*x^(11/3)-4862*a^10*b*x^(10/3)+5720*a^9*b^2*x^3-6864*a^8*b^3*x^(8/3)+8448*a^
7*b^4*x^(7/3)-10752*a^6*b^5*x^2+14336*a^5*b^6*x^(5/3)-20480*a^4*b^7*x^(4/3)+32768*a^3*b^8*x-65536*a^2*b^9*x^(2
/3)+262144*a*b^10*x^(1/3)+524288*b^11)/(b*x^(2/3)+a*x)^(3/2)/a^12

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2566 vs. \(2 (252) = 504\).
time = 197.19, size = 2566, normalized size = 7.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29393*((6442450944*a^3*b^19 + 5368709120*a^3*b^18 - 2013265920*a^3*b^17 - 6113744*a^18 + 402653184*(17*a^6
- 3*a^3)*b^16 + 8388608*(464*a^6 + 53*a^3)*b^15 - 12582912*(246*a^6 + a^3)*b^14 + 1572864*(1036*a^9 - 2560*a^6
- 3*a^3)*b^13 - 524288*(758*a^9 - 1569*a^6)*b^12 - 393216*(5803*a^9 + 124*a^6)*b^11 + 98304*(1315*a^12 - 2092
4*a^9 - 33*a^6)*b^10 - 57344*(2264*a^12 - 3153*a^9)*b^9 - 6144*(83789*a^12 + 2066*a^9)*b^8 - 1536*(46256*a^15
- 159272*a^12 - 267*a^9)*b^7 - 128*(264488*a^15 + 382229*a^12)*b^6 + 9984*(15547*a^15 + 482*a^12)*b^5 - 24*(23
76192*a^18 + 4735792*a^15 + 7887*a^12)*b^4 - 1664*(107856*a^18 - 16759*a^15)*b^3 - 156*(935424*a^18 + 17935*a^
15)*b^2 + 663*(97664*a^18 + 123*a^15)*b)*x^2 + (6442450944*b^22 + 5368709120*b^21 + 402653184*(17*a^3 - 3)*b^1
9 - 2013265920*b^20 + 8388608*(464*a^3 + 53)*b^18 - 6113744*a^15*b^3 - 12582912*(246*a^3 + 1)*b^17 + 1572864*(
1036*a^6 - 2560*a^3 - 3)*b^16 - 524288*(758*a^6 - 1569*a^3)*b^15 - 393216*(5803*a^6 + 124*a^3)*b^14 + 98304*(1
315*a^9 - 20924*a^6 - 33*a^3)*b^13 - 57344*(2264*a^9 - 3153*a^6)*b^12 - 6144*(83789*a^9 + 2066*a^6)*b^11 - 153
6*(46256*a^12 - 159272*a^9 - 267*a^6)*b^10 - 128*(264488*a^12 + 382229*a^9)*b^9 + 9984*(15547*a^12 + 482*a^9)*
b^8 - 24*(2376192*a^15 + 4735792*a^12 + 7887*a^9)*b^7 - 1664*(107856*a^15 - 16759*a^12)*b^6 - 156*(935424*a^15
+ 17935*a^12)*b^5 + 663*(97664*a^15 + 123*a^12)*b^4)*x - 2*(4199*(4096*a^13*b^9 + 6144*a^13*b^8 + 768*a^13*b^
7 - 4096*a^19 - 144*a^16*b^2 + 216*a^16*b - 27*a^16 + 256*(16*a^16 - 7*a^13)*b^6 + 48*(128*a^16 - 3*a^13)*b^5
+ 24*(32*a^16 + 9*a^13)*b^4 - (5888*a^16 + 27*a^13)*b^3)*x^5 - 17446*(4096*a^10*b^12 + 6144*a^10*b^11 + 768*a^
10*b^10 - 144*a^13*b^5 + 216*a^13*b^4 + 256*(16*a^13 - 7*a^10)*b^9 + 48*(128*a^13 - 3*a^10)*b^8 + 24*(32*a^13
+ 9*a^10)*b^7 - (5888*a^13 + 27*a^10)*b^6 - (4096*a^16 + 27*a^13)*b^3)*x^4 + 33536*(4096*a^7*b^15 + 6144*a^7*b
^14 + 768*a^7*b^13 - 144*a^10*b^8 + 216*a^10*b^7 + 256*(16*a^10 - 7*a^7)*b^12 + 48*(128*a^10 - 3*a^7)*b^11 + 2
4*(32*a^10 + 9*a^7)*b^10 - (5888*a^10 + 27*a^7)*b^9 - (4096*a^13 + 27*a^10)*b^6)*x^3 - 118784*(4096*a^4*b^18 +
6144*a^4*b^17 + 768*a^4*b^16 - 144*a^7*b^11 + 216*a^7*b^10 + 256*(16*a^7 - 7*a^4)*b^15 + 48*(128*a^7 - 3*a^4)
*b^14 + 24*(32*a^7 + 9*a^4)*b^13 - (5888*a^7 + 27*a^4)*b^12 - (4096*a^10 + 27*a^7)*b^9)*x^2 - 262144*(4096*a*b
^21 + 6144*a*b^20 + 768*a*b^19 + 256*(16*a^4 - 7*a)*b^18 - 144*a^4*b^14 + 48*(128*a^4 - 3*a)*b^17 + 216*a^4*b^
13 + 24*(32*a^4 + 9*a)*b^16 - (5888*a^4 + 27*a)*b^15 - (4096*a^7 + 27*a^4)*b^12)*x + (2147483648*b^22 + 322122
5472*b^21 + 134217728*(16*a^3 - 7)*b^19 + 402653184*b^20 + 25165824*(128*a^3 - 3)*b^18 - 75497472*a^3*b^15 + 1
2582912*(32*a^3 + 9)*b^17 + 113246208*a^3*b^14 - 524288*(5888*a^3 + 27)*b^16 - 524288*(4096*a^6 + 27*a^3)*b^13
- 9061*(4096*a^12*b^10 + 6144*a^12*b^9 + 768*a^12*b^8 - 144*a^15*b^3 + 216*a^15*b^2 + 256*(16*a^15 - 7*a^12)*
b^7 + 48*(128*a^15 - 3*a^12)*b^6 + 24*(32*a^15 + 9*a^12)*b^5 - (5888*a^15 + 27*a^12)*b^4 - (4096*a^18 + 27*a^1
5)*b)*x^4 + 21032*(4096*a^9*b^13 + 6144*a^9*b^12 + 768*a^9*b^11 - 144*a^12*b^6 + 216*a^12*b^5 + 256*(16*a^12 -
7*a^9)*b^10 + 48*(128*a^12 - 3*a^9)*b^9 + 24*(32*a^12 + 9*a^9)*b^8 - (5888*a^12 + 27*a^9)*b^7 - (4096*a^15 +
27*a^12)*b^4)*x^3 - 45568*(4096*a^6*b^16 + 6144*a^6*b^15 + 768*a^6*b^14 - 144*a^9*b^9 + 216*a^9*b^8 + 256*(16*
a^9 - 7*a^6)*b^13 + 48*(128*a^9 - 3*a^6)*b^12 + 24*(32*a^9 + 9*a^6)*b^11 - (5888*a^9 + 27*a^6)*b^10 - (4096*a^
12 + 27*a^9)*b^7)*x^2 + 360448*(4096*a^3*b^19 + 6144*a^3*b^18 + 768*a^3*b^17 - 144*a^6*b^12 + 216*a^6*b^11 + 2
56*(16*a^6 - 7*a^3)*b^16 + 48*(128*a^6 - 3*a^3)*b^15 + 24*(32*a^6 + 9*a^3)*b^14 - (5888*a^6 + 27*a^3)*b^13 - (
4096*a^9 + 27*a^6)*b^10)*x)*x^(2/3) + 3*(4927*(4096*a^11*b^11 + 6144*a^11*b^10 + 768*a^11*b^9 - 144*a^14*b^4 +
216*a^14*b^3 + 256*(16*a^14 - 7*a^11)*b^8 + 48*(128*a^14 - 3*a^11)*b^7 + 24*(32*a^14 + 9*a^11)*b^6 - (5888*a^
14 + 27*a^11)*b^5 - (4096*a^17 + 27*a^14)*b^2)*x^4 - 8688*(4096*a^8*b^14 + 6144*a^8*b^13 + 768*a^8*b^12 - 144*
a^11*b^7 + 216*a^11*b^6 + 256*(16*a^11 - 7*a^8)*b^11 + 48*(128*a^11 - 3*a^8)*b^10 + 24*(32*a^11 + 9*a^8)*b^9 -
(5888*a^11 + 27*a^8)*b^8 - (4096*a^14 + 27*a^11)*b^5)*x^3 + 22528*(4096*a^5*b^17 + 6144*a^5*b^16 + 768*a^5*b^
15 - 144*a^8*b^10 + 216*a^8*b^9 + 256*(16*a^8 - 7*a^5)*b^14 + 48*(128*a^8 - 3*a^5)*b^13 + 24*(32*a^8 + 9*a^5)*
b^12 - (5888*a^8 + 27*a^5)*b^11 - (4096*a^11 + 27*a^8)*b^8)*x^2 + 65536*(4096*a^2*b^20 + 6144*a^2*b^19 + 768*a
^2*b^18 - 144*a^5*b^13 + 256*(16*a^5 - 7*a^2)*b^17 + 216*a^5*b^12 + 48*(128*a^5 - 3*a^2)*b^16 + 24*(32*a^5 + 9
*a^2)*b^15 - (5888*a^5 + 27*a^2)*b^14 - (4096*a^8 + 27*a^5)*b^11)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((4096*a^
15*b^9 + 6144*a^15*b^8 + 768*a^15*b^7 - 4096*a^21 - 144*a^18*b^2 + 216*a^18*b - 27*a^18 + 256*(16*a^18 - 7*a^1
5)*b^6 + 48*(128*a^18 - 3*a^15)*b^5 + 24*(32*a^18 + 9*a^15)*b^4 - (5888*a^18 + 27*a^15)*b^3)*x^2 + (4096*a^12*
b^12 + 6144*a^12*b^11 + 768*a^12*b^10 - 144*a^15*b^5 + 216*a^15*b^4 + 256*(16*a^15 - 7*a^12)*b^9 + 48*(128*a^1
5 - 3*a^12)*b^8 + 24*(32*a^15 + 9*a^12)*b^7 - (...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.09, size = 214, normalized size = 0.64 \begin {gather*} -\frac {1048576 \, b^{\frac {21}{2}}}{29393 \, a^{12}} + \frac {6 \, b^{11}}{\sqrt {a x^{\frac {1}{3}} + b} a^{12}} + \frac {2 \, {\left (4199 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{240} - 51051 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{240} b + 285285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{240} b^{2} - 969969 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{240} b^{3} + 2238390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{240} b^{4} - 3703518 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{240} b^{5} + 4526522 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{240} b^{6} - 4157010 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{240} b^{7} + 2909907 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{240} b^{8} - 1616615 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{240} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} a^{240} b^{10}\right )}}{29393 \, a^{252}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1048576/29393*b^(21/2)/a^12 + 6*b^11/(sqrt(a*x^(1/3) + b)*a^12) + 2/29393*(4199*(a*x^(1/3) + b)^(21/2)*a^240
- 51051*(a*x^(1/3) + b)^(19/2)*a^240*b + 285285*(a*x^(1/3) + b)^(17/2)*a^240*b^2 - 969969*(a*x^(1/3) + b)^(15/
2)*a^240*b^3 + 2238390*(a*x^(1/3) + b)^(13/2)*a^240*b^4 - 3703518*(a*x^(1/3) + b)^(11/2)*a^240*b^5 + 4526522*(
a*x^(1/3) + b)^(9/2)*a^240*b^6 - 4157010*(a*x^(1/3) + b)^(7/2)*a^240*b^7 + 2909907*(a*x^(1/3) + b)^(5/2)*a^240
*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*a^240*b^9 + 969969*sqrt(a*x^(1/3) + b)*a^240*b^10)/a^252

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a*x + b*x^(2/3))^(3/2),x)

[Out]

int(x^4/(a*x + b*x^(2/3))^(3/2), x)

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