3.2.16 \(\int \frac {x^3}{(b x^{2/3}+a x)^{3/2}} \, dx\)

Optimal. Leaf size=248 \[ -\frac {65536 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^9 \sqrt [3]{x}}+\frac {32768 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^8}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^6}-\frac {3584 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^3}+\frac {32 x^2 \sqrt {a x+b x^{2/3}}}{5 a^2}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.41, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2015, 2016, 2002, 2014} \begin {gather*} -\frac {65536 b^7 \sqrt {a x+b x^{2/3}}}{2145 a^9 \sqrt [3]{x}}+\frac {32768 b^6 \sqrt {a x+b x^{2/3}}}{2145 a^8}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {a x+b x^{2/3}}}{429 a^6}-\frac {3584 b^3 x \sqrt {a x+b x^{2/3}}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {a x+b x^{2/3}}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {a x+b x^{2/3}}}{65 a^3}+\frac {32 x^2 \sqrt {a x+b x^{2/3}}}{5 a^2}-\frac {6 x^3}{a \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^3)/(a*Sqrt[b*x^(2/3) + a*x]) + (32768*b^6*Sqrt[b*x^(2/3) + a*x])/(2145*a^8) - (65536*b^7*Sqrt[b*x^(2/3)
+ a*x])/(2145*a^9*x^(1/3)) - (8192*b^5*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(715*a^7) + (4096*b^4*x^(2/3)*Sqrt[b*x^(
2/3) + a*x])/(429*a^6) - (3584*b^3*x*Sqrt[b*x^(2/3) + a*x])/(429*a^5) + (5376*b^2*x^(4/3)*Sqrt[b*x^(2/3) + a*x
])/(715*a^4) - (448*b*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(65*a^3) + (32*x^2*Sqrt[b*x^(2/3) + a*x])/(5*a^2)

Rule 2002

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 2014

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] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2015

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 2016

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^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {16 \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{a}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}-\frac {(224 b) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{15 a^2}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}+\frac {\left (896 b^2\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{65 a^3}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}-\frac {\left (1792 b^3\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{143 a^4}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}+\frac {\left (14336 b^4\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{1287 a^5}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {4096 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^6}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}-\frac {\left (4096 b^5\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{429 a^6}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^6}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}+\frac {\left (16384 b^6\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{2145 a^7}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {32768 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^8}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^6}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}-\frac {\left (32768 b^7\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{6435 a^8}\\ &=-\frac {6 x^3}{a \sqrt {b x^{2/3}+a x}}+\frac {32768 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^8}-\frac {65536 b^7 \sqrt {b x^{2/3}+a x}}{2145 a^9 \sqrt [3]{x}}-\frac {8192 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^7}+\frac {4096 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^6}-\frac {3584 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^5}+\frac {5376 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^4}-\frac {448 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^3}+\frac {32 x^2 \sqrt {b x^{2/3}+a x}}{5 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 122, normalized size = 0.49 \begin {gather*} \frac {2 \left (429 a^8 x^3-528 a^7 b x^{8/3}+672 a^6 b^2 x^{7/3}-896 a^5 b^3 x^2+1280 a^4 b^4 x^{5/3}-2048 a^3 b^5 x^{4/3}+4096 a^2 b^6 x-16384 a b^7 x^{2/3}-32768 b^8 \sqrt [3]{x}\right )}{2145 a^9 \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-32768*b^8*x^(1/3) - 16384*a*b^7*x^(2/3) + 4096*a^2*b^6*x - 2048*a^3*b^5*x^(4/3) + 1280*a^4*b^4*x^(5/3) -
896*a^5*b^3*x^2 + 672*a^6*b^2*x^(7/3) - 528*a^7*b*x^(8/3) + 429*a^8*x^3))/(2145*a^9*Sqrt[b*x^(2/3) + a*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.94, size = 128, normalized size = 0.52 \begin {gather*} -\frac {2 \sqrt [3]{x} \left (-429 a^8 x^{8/3}+528 a^7 b x^{7/3}-672 a^6 b^2 x^2+896 a^5 b^3 x^{5/3}-1280 a^4 b^4 x^{4/3}+2048 a^3 b^5 x-4096 a^2 b^6 x^{2/3}+16384 a b^7 \sqrt [3]{x}+32768 b^8\right )}{2145 a^9 \sqrt {x^{2/3} \left (a \sqrt [3]{x}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^(1/3)*(32768*b^8 + 16384*a*b^7*x^(1/3) - 4096*a^2*b^6*x^(2/3) + 2048*a^3*b^5*x - 1280*a^4*b^4*x^(4/3) +
896*a^5*b^3*x^(5/3) - 672*a^6*b^2*x^2 + 528*a^7*b*x^(7/3) - 429*a^8*x^(8/3)))/(2145*a^9*Sqrt[(b + a*x^(1/3))*x
^(2/3)])

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 0.21, size = 163, normalized size = 0.66 \begin {gather*} \frac {65536 \, b^{\frac {15}{2}}}{2145 \, a^{9}} - \frac {6 \, b^{8}}{\sqrt {a x^{\frac {1}{3}} + b} a^{9}} + \frac {2 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{126} - 3960 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{126} b + 16380 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{126} b^{2} - 40040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{126} b^{3} + 64350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{126} b^{4} - 72072 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{126} b^{5} + 60060 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{126} b^{6} - 51480 \, \sqrt {a x^{\frac {1}{3}} + b} a^{126} b^{7}\right )}}{2145 \, a^{135}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

65536/2145*b^(15/2)/a^9 - 6*b^8/(sqrt(a*x^(1/3) + b)*a^9) + 2/2145*(429*(a*x^(1/3) + b)^(15/2)*a^126 - 3960*(a
*x^(1/3) + b)^(13/2)*a^126*b + 16380*(a*x^(1/3) + b)^(11/2)*a^126*b^2 - 40040*(a*x^(1/3) + b)^(9/2)*a^126*b^3
+ 64350*(a*x^(1/3) + b)^(7/2)*a^126*b^4 - 72072*(a*x^(1/3) + b)^(5/2)*a^126*b^5 + 60060*(a*x^(1/3) + b)^(3/2)*
a^126*b^6 - 51480*sqrt(a*x^(1/3) + b)*a^126*b^7)/a^135

________________________________________________________________________________________

maple [A]  time = 0.05, size = 110, normalized size = 0.44 \begin {gather*} \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (429 a^{8} x^{\frac {8}{3}}-528 a^{7} b \,x^{\frac {7}{3}}+672 a^{6} b^{2} x^{2}-896 a^{5} b^{3} x^{\frac {5}{3}}+1280 a^{4} b^{4} x^{\frac {4}{3}}-2048 a^{3} b^{5} x +4096 a^{2} b^{6} x^{\frac {2}{3}}-16384 a \,b^{7} x^{\frac {1}{3}}-32768 b^{8}\right ) x}{2145 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} a^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2145*x*(a*x^(1/3)+b)*(429*a^8*x^(8/3)-528*a^7*b*x^(7/3)+672*a^6*b^2*x^2-896*a^5*b^3*x^(5/3)+1280*x^(4/3)*a^4
*b^4-2048*a^3*b^5*x+4096*x^(2/3)*a^2*b^6-16384*x^(1/3)*a*b^7-32768*b^8)/(a*x+b*x^(2/3))^(3/2)/a^9

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

int(x^3/(a*x + b*x^(2/3))^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\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**3/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

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

________________________________________________________________________________________