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

Optimal. Leaf size=313 \[ \frac {524288 b^{10} \sqrt {a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac {262144 b^9 \sqrt {a x+b x^{2/3}}}{323323 a^{10}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {a x+b x^{2/3}}}{323323 a^8}+\frac {20480 b^6 x \sqrt {a x+b x^{2/3}}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {a x+b x^{2/3}}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {a x+b x^{2/3}}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {a x+b x^{2/3}}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {a x+b x^{2/3}}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {a x+b x^{2/3}}}{133 a^2}+\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a} \]

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Rubi [A]  time = 0.53, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \begin {gather*} \frac {524288 b^{10} \sqrt {a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac {262144 b^9 \sqrt {a x+b x^{2/3}}}{323323 a^{10}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {a x+b x^{2/3}}}{323323 a^8}+\frac {20480 b^6 x \sqrt {a x+b x^{2/3}}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {a x+b x^{2/3}}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {a x+b x^{2/3}}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {a x+b x^{2/3}}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {a x+b x^{2/3}}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {a x+b x^{2/3}}}{133 a^2}+\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[b*x^(2/3) + a*x],x]

[Out]

(-262144*b^9*Sqrt[b*x^(2/3) + a*x])/(323323*a^10) + (524288*b^10*Sqrt[b*x^(2/3) + a*x])/(323323*a^11*x^(1/3))
+ (196608*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(323323*a^9) - (163840*b^7*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(323323
*a^8) + (20480*b^6*x*Sqrt[b*x^(2/3) + a*x])/(46189*a^7) - (18432*b^5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(46189*a^6
) + (1536*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^5) - (768*b^3*x^2*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) + (72
0*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^3) - (40*b*x^(8/3)*Sqrt[b*x^(2/3) + a*x])/(133*a^2) + (2*x^3*Sqrt
[b*x^(2/3) + a*x])/(7*a)

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 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}{\sqrt {b x^{2/3}+a x}} \, dx &=\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {(20 b) \int \frac {x^{8/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{21 a}\\ &=-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (120 b^2\right ) \int \frac {x^{7/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{133 a^2}\\ &=\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (1920 b^3\right ) \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx}{2261 a^3}\\ &=-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (256 b^4\right ) \int \frac {x^{5/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{323 a^4}\\ &=\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (3072 b^5\right ) \int \frac {x^{4/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{4199 a^5}\\ &=-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (30720 b^6\right ) \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx}{46189 a^6}\\ &=\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (81920 b^7\right ) \int \frac {x^{2/3}}{\sqrt {b x^{2/3}+a x}} \, dx}{138567 a^7}\\ &=-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (163840 b^8\right ) \int \frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}} \, dx}{323323 a^8}\\ &=\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}-\frac {\left (131072 b^9\right ) \int \frac {1}{\sqrt {b x^{2/3}+a x}} \, dx}{323323 a^9}\\ &=-\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}+\frac {\left (262144 b^{10}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt {b x^{2/3}+a x}} \, dx}{969969 a^{10}}\\ &=-\frac {262144 b^9 \sqrt {b x^{2/3}+a x}}{323323 a^{10}}+\frac {524288 b^{10} \sqrt {b x^{2/3}+a x}}{323323 a^{11} \sqrt [3]{x}}+\frac {196608 b^8 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{323323 a^9}-\frac {163840 b^7 x^{2/3} \sqrt {b x^{2/3}+a x}}{323323 a^8}+\frac {20480 b^6 x \sqrt {b x^{2/3}+a x}}{46189 a^7}-\frac {18432 b^5 x^{4/3} \sqrt {b x^{2/3}+a x}}{46189 a^6}+\frac {1536 b^4 x^{5/3} \sqrt {b x^{2/3}+a x}}{4199 a^5}-\frac {768 b^3 x^2 \sqrt {b x^{2/3}+a x}}{2261 a^4}+\frac {720 b^2 x^{7/3} \sqrt {b x^{2/3}+a x}}{2261 a^3}-\frac {40 b x^{8/3} \sqrt {b x^{2/3}+a x}}{133 a^2}+\frac {2 x^3 \sqrt {b x^{2/3}+a x}}{7 a}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 148, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {a x+b x^{2/3}} \left (46189 a^{10} x^{10/3}-48620 a^9 b x^3+51480 a^8 b^2 x^{8/3}-54912 a^7 b^3 x^{7/3}+59136 a^6 b^4 x^2-64512 a^5 b^5 x^{5/3}+71680 a^4 b^6 x^{4/3}-81920 a^3 b^7 x+98304 a^2 b^8 x^{2/3}-131072 a b^9 \sqrt [3]{x}+262144 b^{10}\right )}{323323 a^{11} \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(262144*b^10 - 131072*a*b^9*x^(1/3) + 98304*a^2*b^8*x^(2/3) - 81920*a^3*b^7*x + 71680
*a^4*b^6*x^(4/3) - 64512*a^5*b^5*x^(5/3) + 59136*a^6*b^4*x^2 - 54912*a^7*b^3*x^(7/3) + 51480*a^8*b^2*x^(8/3) -
48620*a^9*b*x^3 + 46189*a^10*x^(10/3)))/(323323*a^11*x^(1/3))

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IntegrateAlgebraic [A]  time = 0.11, size = 148, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {a x+b x^{2/3}} \left (46189 a^{10} x^{10/3}-48620 a^9 b x^3+51480 a^8 b^2 x^{8/3}-54912 a^7 b^3 x^{7/3}+59136 a^6 b^4 x^2-64512 a^5 b^5 x^{5/3}+71680 a^4 b^6 x^{4/3}-81920 a^3 b^7 x+98304 a^2 b^8 x^{2/3}-131072 a b^9 \sqrt [3]{x}+262144 b^{10}\right )}{323323 a^{11} \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(262144*b^10 - 131072*a*b^9*x^(1/3) + 98304*a^2*b^8*x^(2/3) - 81920*a^3*b^7*x + 71680
*a^4*b^6*x^(4/3) - 64512*a^5*b^5*x^(5/3) + 59136*a^6*b^4*x^2 - 54912*a^7*b^3*x^(7/3) + 51480*a^8*b^2*x^(8/3) -
48620*a^9*b*x^3 + 46189*a^10*x^(10/3)))/(323323*a^11*x^(1/3))

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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)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.19, size = 164, normalized size = 0.52 \begin {gather*} -\frac {524288 \, b^{\frac {21}{2}}}{323323 \, a^{11}} + \frac {2 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{323323 \, a^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")

[Out]

-524288/323323*b^(21/2)/a^11 + 2/323323*(46189*(a*x^(1/3) + b)^(21/2) - 510510*(a*x^(1/3) + b)^(19/2)*b + 2567
565*(a*x^(1/3) + b)^(17/2)*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*b^3 + 15668730*(a*x^(1/3) + b)^(13/2)*b^4 - 22
221108*(a*x^(1/3) + b)^(11/2)*b^5 + 22632610*(a*x^(1/3) + b)^(9/2)*b^6 - 16628040*(a*x^(1/3) + b)^(7/2)*b^7 +
8729721*(a*x^(1/3) + b)^(5/2)*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*b^9 + 969969*sqrt(a*x^(1/3) + b)*b^10)/a^11

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maple [A]  time = 0.04, size = 134, normalized size = 0.43 \begin {gather*} \frac {2 \left (a \,x^{\frac {1}{3}}+b \right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 a^{6} b^{4} x^{2}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} b^{6} x^{\frac {4}{3}}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right ) x^{\frac {1}{3}}}{323323 \sqrt {a x +b \,x^{\frac {2}{3}}}\, a^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x+b*x^(2/3))^(1/2),x)

[Out]

2/323323*x^(1/3)*(a*x^(1/3)+b)*(46189*x^(10/3)*a^10-48620*x^3*a^9*b+51480*a^8*b^2*x^(8/3)-54912*x^(7/3)*a^7*b^
3+59136*a^6*b^4*x^2-64512*a^5*b^5*x^(5/3)+71680*x^(4/3)*a^4*b^6-81920*x*a^3*b^7+98304*a^2*b^8*x^(2/3)-131072*x
^(1/3)*a*b^9+262144*b^10)/(a*x+b*x^(2/3))^(1/2)/a^11

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x + b*x^(2/3))^(1/2),x)

[Out]

int(x^3/(a*x + b*x^(2/3))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**(2/3)+a*x)**(1/2),x)

[Out]

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

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