3.175 \(\int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx\)

Optimal. Leaf size=354 \[ -\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{262144 b^{21/2}}+\frac {12597 a^{10} \sqrt {a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac {4199 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^9 x}+\frac {4199 a^8 \sqrt {a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac {12597 a^7 \sqrt {a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac {4199 a^6 \sqrt {a x+b x^{2/3}}}{215040 b^6 x^2}-\frac {4199 a^5 \sqrt {a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac {323 a^4 \sqrt {a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{21120 b^3 x^3}+\frac {19 a^2 \sqrt {a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{220 b x^{11/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4} \]

[Out]

-12597/262144*a^11*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(21/2)-3/11*(b*x^(2/3)+a*x)^(1/2)/x^4-3/22
0*a*(b*x^(2/3)+a*x)^(1/2)/b/x^(11/3)+19/1320*a^2*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(10/3)-323/21120*a^3*(b*x^(2/3)+a
*x)^(1/2)/b^3/x^3+323/19712*a^4*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(8/3)-4199/236544*a^5*(b*x^(2/3)+a*x)^(1/2)/b^5/x^
(7/3)+4199/215040*a^6*(b*x^(2/3)+a*x)^(1/2)/b^6/x^2-12597/573440*a^7*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(5/3)+4199/16
3840*a^8*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/3)-4199/131072*a^9*(b*x^(2/3)+a*x)^(1/2)/b^9/x+12597/262144*a^10*(b*x^
(2/3)+a*x)^(1/2)/b^10/x^(2/3)

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Rubi [A]  time = 0.66, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2029, 206} \[ \frac {12597 a^{10} \sqrt {a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac {4199 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^9 x}+\frac {4199 a^8 \sqrt {a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac {12597 a^7 \sqrt {a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac {4199 a^6 \sqrt {a x+b x^{2/3}}}{215040 b^6 x^2}-\frac {4199 a^5 \sqrt {a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac {323 a^4 \sqrt {a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{21120 b^3 x^3}+\frac {19 a^2 \sqrt {a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{262144 b^{21/2}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{220 b x^{11/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(2/3) + a*x]/x^5,x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) - (3*a*Sqrt[b*x^(2/3) + a*x])/(220*b*x^(11/3)) + (19*a^2*Sqrt[b*x^(2/3) +
a*x])/(1320*b^2*x^(10/3)) - (323*a^3*Sqrt[b*x^(2/3) + a*x])/(21120*b^3*x^3) + (323*a^4*Sqrt[b*x^(2/3) + a*x])/
(19712*b^4*x^(8/3)) - (4199*a^5*Sqrt[b*x^(2/3) + a*x])/(236544*b^5*x^(7/3)) + (4199*a^6*Sqrt[b*x^(2/3) + a*x])
/(215040*b^6*x^2) - (12597*a^7*Sqrt[b*x^(2/3) + a*x])/(573440*b^7*x^(5/3)) + (4199*a^8*Sqrt[b*x^(2/3) + a*x])/
(163840*b^8*x^(4/3)) - (4199*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^9*x) + (12597*a^10*Sqrt[b*x^(2/3) + a*x])/(2
62144*b^10*x^(2/3)) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(262144*b^(21/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2020

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

Rule 2025

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2029

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}+\frac {1}{22} a \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}-\frac {\left (19 a^2\right ) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{440 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}+\frac {\left (323 a^3\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{7920 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}-\frac {\left (323 a^4\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{8448 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}+\frac {\left (4199 a^5\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{118272 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}-\frac {\left (4199 a^6\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{129024 b^5}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}+\frac {\left (4199 a^7\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{143360 b^6}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}-\frac {\left (4199 a^8\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{163840 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}+\frac {\left (4199 a^9\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{196608 b^8}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}-\frac {\left (4199 a^{10}\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^9}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}+\frac {\left (4199 a^{11}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{524288 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {\left (12597 a^{11}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{10}}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {12597 a^{11} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 57, normalized size = 0.16 \[ \frac {2 a^{11} \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {3}{2},12;\frac {5}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^{12} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^(2/3) + a*x]/x^5,x]

[Out]

(2*a^11*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[3/2, 12, 5/2, 1 + (a*x^(1/3))/b])/(b^12*x^(1/3
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^5,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.43, size = 228, normalized size = 0.64 \[ \frac {\frac {14549535 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} - 155195040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b + 749786037 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{2} - 2163862272 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{3} + 4139920070 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{4} - 5503713280 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{5} + 5174056250 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{6} - 3424523520 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{7} + 1551313995 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{8} - 450357600 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{9} - 14549535 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{10}}{a^{11} b^{10} x^{\frac {11}{3}}}}{302776320 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/302776320*(14549535*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(2
1/2)*a^12 - 155195040*(a*x^(1/3) + b)^(19/2)*a^12*b + 749786037*(a*x^(1/3) + b)^(17/2)*a^12*b^2 - 2163862272*(
a*x^(1/3) + b)^(15/2)*a^12*b^3 + 4139920070*(a*x^(1/3) + b)^(13/2)*a^12*b^4 - 5503713280*(a*x^(1/3) + b)^(11/2
)*a^12*b^5 + 5174056250*(a*x^(1/3) + b)^(9/2)*a^12*b^6 - 3424523520*(a*x^(1/3) + b)^(7/2)*a^12*b^7 + 155131399
5*(a*x^(1/3) + b)^(5/2)*a^12*b^8 - 450357600*(a*x^(1/3) + b)^(3/2)*a^12*b^9 - 14549535*sqrt(a*x^(1/3) + b)*a^1
2*b^10)/(a^11*b^10*x^(11/3)))/a

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maple [A]  time = 0.06, size = 209, normalized size = 0.59 \[ -\frac {\sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (14549535 a^{11} b^{10} x^{\frac {11}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )[phone] \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {41}{2}}[phone] \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {39}{2}}-1551313995 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {5}{2}} b^{\frac {37}{2}}[phone] \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {7}{2}} b^{\frac {35}{2}}-5174056250 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {9}{2}} b^{\frac {33}{2}}[phone] \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {11}{2}} b^{\frac {31}{2}}-4139920070 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {13}{2}} b^{\frac {29}{2}}[phone] \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {15}{2}} b^{\frac {27}{2}}-749786037 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {17}{2}} b^{\frac {25}{2}}[phone] \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {19}{2}} b^{\frac {23}{2}}-14549535 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {21}{2}} b^{\frac {21}{2}}\right )}{302776320 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {41}{2}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b*x^(2/3))^(1/2)/x^5,x)

[Out]

-1/302776320*(a*x+b*x^(2/3))^(1/2)*(-14549535*(a*x^(1/3)+b)^(21/2)*b^(21/2)[phone]*(a*x^(1/3)+b)^(19/2)*b^(
23/2)-749786037*(a*x^(1/3)+b)^(17/2)*b^(25/2)[phone]*(a*x^(1/3)+b)^(15/2)*b^(27/2)-4139920070*(a*x^(1/3)+b
)^(13/2)*b^(29/2)[phone]*(a*x^(1/3)+b)^(11/2)*b^(31/2)-5174056250*(a*x^(1/3)+b)^(9/2)*b^(33/2)[phone]*
(a*x^(1/3)+b)^(7/2)*b^(35/2)-1551313995*(a*x^(1/3)+b)^(5/2)*b^(37/2)[phone]*arctanh((a*x^(1/3)+b)^(1/2)/b^(1
/2))*b^10*x^(11/3)*a^11+450357600*(a*x^(1/3)+b)^(3/2)*b^(39/2)[phone]*(a*x^(1/3)+b)^(1/2)*b^(41/2))/x^4/(a*x
^(1/3)+b)^(1/2)/b^(41/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(2/3))/x^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^(2/3))^(1/2)/x^5,x)

[Out]

int((a*x + b*x^(2/3))^(1/2)/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**(2/3)+a*x)**(1/2)/x**5,x)

[Out]

Integral(sqrt(a*x + b*x**(2/3))/x**5, x)

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