3.2.86 \(\int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx\) [186]

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

[Out]

-262144/323323*b^9*(b*x^(2/3)+a*x)^(1/2)/a^10+524288/323323*b^10*(b*x^(2/3)+a*x)^(1/2)/a^11/x^(1/3)+196608/323
323*b^8*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^9-163840/323323*b^7*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^8+20480/46189*b^6*
x*(b*x^(2/3)+a*x)^(1/2)/a^7-18432/46189*b^5*x^(4/3)*(b*x^(2/3)+a*x)^(1/2)/a^6+1536/4199*b^4*x^(5/3)*(b*x^(2/3)
+a*x)^(1/2)/a^5-768/2261*b^3*x^2*(b*x^(2/3)+a*x)^(1/2)/a^4+720/2261*b^2*x^(7/3)*(b*x^(2/3)+a*x)^(1/2)/a^3-40/1
33*b*x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a^2+2/7*x^3*(b*x^(2/3)+a*x)^(1/2)/a

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Rubi [A]
time = 0.36, 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 = {2041, 2027, 2039} \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 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 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^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 = 0.09, size = 148, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {b x^{2/3}+a x} \left (262144 b^{10}-131072 a b^9 \sqrt [3]{x}+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}\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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Maple [A]
time = 0.36, size = 134, normalized size = 0.43

method	result	size
derivativedivides	\(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} b^{6} x^{\frac {4}{3}}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\)	\(134\)
default	\(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (46189 a^{10} x^{\frac {10}{3}}-48620 a^{9} b \,x^{3}+51480 a^{8} b^{2} x^{\frac {8}{3}}-54912 a^{7} b^{3} x^{\frac {7}{3}}+59136 x^{2} a^{6} b^{4}-64512 a^{5} b^{5} x^{\frac {5}{3}}+71680 a^{4} b^{6} x^{\frac {4}{3}}-81920 a^{3} b^{7} x +98304 a^{2} b^{8} x^{\frac {2}{3}}-131072 a \,b^{9} x^{\frac {1}{3}}+262144 b^{10}\right )}{323323 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{11}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/323323*x^(1/3)*(b+a*x^(1/3))*(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*x^2*a^6*b^4-64512*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*x^(1/3)+262144*b^10)/(b*x^(2/3)+a*x)^(1/2)/a^11

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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^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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1031 vs. \(2 (233) = 466\).
time = 242.12, size = 1031, normalized size = 3.29 \begin {gather*} -\frac {2 \, {\left ({\left (3298534883328 \, b^{16} + 687194767360 \, b^{15} + 3221225472 \, {\left (64 \, a^{3} - 3\right )} b^{13} - 64424509440 \, b^{14} - 16777216 \, {\left (11264 \, a^{3} - 53\right )} b^{12} - 269004736 \, a^{12} - 6291456 \, {\left (5504 \, a^{3} + 1\right )} b^{11} + 196608 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{10} + 7340032 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{9} - 786432 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{8} - 12288 \, {\left (45731840 \, a^{9} - 495872 \, a^{6} - 15 \, a^{3}\right )} b^{7} - 114688 \, {\left (1349120 \, a^{9} + 3439 \, a^{6}\right )} b^{6} + 3913728 \, {\left (5600 \, a^{9} + 3 \, a^{6}\right )} b^{5} - 2112 \, {\left (101384192 \, a^{12} + 1958400 \, a^{9} + 63 \, a^{6}\right )} b^{4} - 36608 \, {\left (3784704 \, a^{12} - 8101 \, a^{9}\right )} b^{3} - 109824 \, {\left (226688 \, a^{12} + 85 \, a^{9}\right )} b^{2} + 7293 \, {\left (974848 \, a^{12} + 15 \, a^{9}\right )} b\right )} x - {\left (46189 \, {\left (16777216 \, a^{10} b^{6} + 6291456 \, a^{10} b^{5} + 196608 \, a^{10} b^{4} - 262144 \, a^{13} - 114688 \, a^{10} b^{3} - 2304 \, a^{10} b^{2} + 864 \, a^{10} b - 27 \, a^{10}\right )} x^{4} - 54912 \, {\left (16777216 \, a^{7} b^{9} + 6291456 \, a^{7} b^{8} + 196608 \, a^{7} b^{7} - 114688 \, a^{7} b^{6} - 2304 \, a^{7} b^{5} + 864 \, a^{7} b^{4} - {\left (262144 \, a^{10} + 27 \, a^{7}\right )} b^{3}\right )} x^{3} + 71680 \, {\left (16777216 \, a^{4} b^{12} + 6291456 \, a^{4} b^{11} + 196608 \, a^{4} b^{10} - 114688 \, a^{4} b^{9} - 2304 \, a^{4} b^{8} + 864 \, a^{4} b^{7} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{6}\right )} x^{2} - 131072 \, {\left (16777216 \, a b^{15} + 6291456 \, a b^{14} + 196608 \, a b^{13} - 114688 \, a b^{12} - 2304 \, a b^{11} + 864 \, a b^{10} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{9}\right )} x + 4 \, {\left (1099511627776 \, b^{16} + 412316860416 \, b^{15} + 12884901888 \, b^{14} - 7516192768 \, b^{13} - 150994944 \, b^{12} - 65536 \, {\left (262144 \, a^{3} + 27\right )} b^{10} + 56623104 \, b^{11} - 12155 \, {\left (16777216 \, a^{9} b^{7} + 6291456 \, a^{9} b^{6} + 196608 \, a^{9} b^{5} - 114688 \, a^{9} b^{4} - 2304 \, a^{9} b^{3} + 864 \, a^{9} b^{2} - {\left (262144 \, a^{12} + 27 \, a^{9}\right )} b\right )} x^{3} + 14784 \, {\left (16777216 \, a^{6} b^{10} + 6291456 \, a^{6} b^{9} + 196608 \, a^{6} b^{8} - 114688 \, a^{6} b^{7} - 2304 \, a^{6} b^{6} + 864 \, a^{6} b^{5} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b^{4}\right )} x^{2} - 20480 \, {\left (16777216 \, a^{3} b^{13} + 6291456 \, a^{3} b^{12} + 196608 \, a^{3} b^{11} - 114688 \, a^{3} b^{10} - 2304 \, a^{3} b^{9} + 864 \, a^{3} b^{8} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{7}\right )} x\right )} x^{\frac {2}{3}} + 24 \, {\left (2145 \, {\left (16777216 \, a^{8} b^{8} + 6291456 \, a^{8} b^{7} + 196608 \, a^{8} b^{6} - 114688 \, a^{8} b^{5} - 2304 \, a^{8} b^{4} + 864 \, a^{8} b^{3} - {\left (262144 \, a^{11} + 27 \, a^{8}\right )} b^{2}\right )} x^{3} - 2688 \, {\left (16777216 \, a^{5} b^{11} + 6291456 \, a^{5} b^{10} + 196608 \, a^{5} b^{9} - 114688 \, a^{5} b^{8} - 2304 \, a^{5} b^{7} + 864 \, a^{5} b^{6} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{5}\right )} x^{2} + 4096 \, {\left (16777216 \, a^{2} b^{14} + 6291456 \, a^{2} b^{13} + 196608 \, a^{2} b^{12} - 114688 \, a^{2} b^{11} - 2304 \, a^{2} b^{10} + 864 \, a^{2} b^{9} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{8}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}\right )}}{323323 \, {\left (16777216 \, a^{11} b^{6} + 6291456 \, a^{11} b^{5} + 196608 \, a^{11} b^{4} - 262144 \, a^{14} - 114688 \, a^{11} b^{3} - 2304 \, a^{11} b^{2} + 864 \, a^{11} b - 27 \, a^{11}\right )} 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="fricas")

[Out]

-2/323323*((3298534883328*b^16 + 687194767360*b^15 + 3221225472*(64*a^3 - 3)*b^13 - 64424509440*b^14 - 1677721
6*(11264*a^3 - 53)*b^12 - 269004736*a^12 - 6291456*(5504*a^3 + 1)*b^11 + 196608*(3194880*a^6 - 114688*a^3 - 3)
*b^10 + 7340032*(18816*a^6 + 103*a^3)*b^9 - 786432*(48816*a^6 + 23*a^3)*b^8 - 12288*(45731840*a^9 - 495872*a^6
- 15*a^3)*b^7 - 114688*(1349120*a^9 + 3439*a^6)*b^6 + 3913728*(5600*a^9 + 3*a^6)*b^5 - 2112*(101384192*a^12 +
1958400*a^9 + 63*a^6)*b^4 - 36608*(3784704*a^12 - 8101*a^9)*b^3 - 109824*(226688*a^12 + 85*a^9)*b^2 + 7293*(9
74848*a^12 + 15*a^9)*b)*x - (46189*(16777216*a^10*b^6 + 6291456*a^10*b^5 + 196608*a^10*b^4 - 262144*a^13 - 114
688*a^10*b^3 - 2304*a^10*b^2 + 864*a^10*b - 27*a^10)*x^4 - 54912*(16777216*a^7*b^9 + 6291456*a^7*b^8 + 196608*
a^7*b^7 - 114688*a^7*b^6 - 2304*a^7*b^5 + 864*a^7*b^4 - (262144*a^10 + 27*a^7)*b^3)*x^3 + 71680*(16777216*a^4*
b^12 + 6291456*a^4*b^11 + 196608*a^4*b^10 - 114688*a^4*b^9 - 2304*a^4*b^8 + 864*a^4*b^7 - (262144*a^7 + 27*a^4
)*b^6)*x^2 - 131072*(16777216*a*b^15 + 6291456*a*b^14 + 196608*a*b^13 - 114688*a*b^12 - 2304*a*b^11 + 864*a*b^
10 - (262144*a^4 + 27*a)*b^9)*x + 4*(1099511627776*b^16 + 412316860416*b^15 + 12884901888*b^14 - 7516192768*b^
13 - 150994944*b^12 - 65536*(262144*a^3 + 27)*b^10 + 56623104*b^11 - 12155*(16777216*a^9*b^7 + 6291456*a^9*b^6
+ 196608*a^9*b^5 - 114688*a^9*b^4 - 2304*a^9*b^3 + 864*a^9*b^2 - (262144*a^12 + 27*a^9)*b)*x^3 + 14784*(16777
216*a^6*b^10 + 6291456*a^6*b^9 + 196608*a^6*b^8 - 114688*a^6*b^7 - 2304*a^6*b^6 + 864*a^6*b^5 - (262144*a^9 +
27*a^6)*b^4)*x^2 - 20480*(16777216*a^3*b^13 + 6291456*a^3*b^12 + 196608*a^3*b^11 - 114688*a^3*b^10 - 2304*a^3*
b^9 + 864*a^3*b^8 - (262144*a^6 + 27*a^3)*b^7)*x)*x^(2/3) + 24*(2145*(16777216*a^8*b^8 + 6291456*a^8*b^7 + 196
608*a^8*b^6 - 114688*a^8*b^5 - 2304*a^8*b^4 + 864*a^8*b^3 - (262144*a^11 + 27*a^8)*b^2)*x^3 - 2688*(16777216*a
^5*b^11 + 6291456*a^5*b^10 + 196608*a^5*b^9 - 114688*a^5*b^8 - 2304*a^5*b^7 + 864*a^5*b^6 - (262144*a^8 + 27*a
^5)*b^5)*x^2 + 4096*(16777216*a^2*b^14 + 6291456*a^2*b^13 + 196608*a^2*b^12 - 114688*a^2*b^11 - 2304*a^2*b^10
+ 864*a^2*b^9 - (262144*a^5 + 27*a^2)*b^8)*x)*x^(1/3))*sqrt(a*x + b*x^(2/3)))/((16777216*a^11*b^6 + 6291456*a^
11*b^5 + 196608*a^11*b^4 - 262144*a^14 - 114688*a^11*b^3 - 2304*a^11*b^2 + 864*a^11*b - 27*a^11)*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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Giac [A]
time = 1.13, 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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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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