3.1.95 \(\int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx\)

Optimal. Leaf size=266 \[ \frac {1287 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{16384 b^{15/2}}-\frac {1287 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^7 x^{2/3}}+\frac {429 a^6 \sqrt {a x+b x^{2/3}}}{8192 b^6 x}-\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{10240 b^5 x^{4/3}}+\frac {1287 a^4 \sqrt {a x+b x^{2/3}}}{35840 b^4 x^{5/3}}-\frac {143 a^3 \sqrt {a x+b x^{2/3}}}{4480 b^3 x^2}+\frac {13 a^2 \sqrt {a x+b x^{2/3}}}{448 b^2 x^{7/3}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{112 b x^{8/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3} \]

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Rubi [A]  time = 0.48, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2029, 206} \begin {gather*} -\frac {1287 a^7 \sqrt {a x+b x^{2/3}}}{16384 b^7 x^{2/3}}+\frac {429 a^6 \sqrt {a x+b x^{2/3}}}{8192 b^6 x}-\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{10240 b^5 x^{4/3}}+\frac {1287 a^4 \sqrt {a x+b x^{2/3}}}{35840 b^4 x^{5/3}}-\frac {143 a^3 \sqrt {a x+b x^{2/3}}}{4480 b^3 x^2}+\frac {13 a^2 \sqrt {a x+b x^{2/3}}}{448 b^2 x^{7/3}}+\frac {1287 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{16384 b^{15/2}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{112 b x^{8/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(2/3) + a*x]/x^4,x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(8*x^3) - (3*a*Sqrt[b*x^(2/3) + a*x])/(112*b*x^(8/3)) + (13*a^2*Sqrt[b*x^(2/3) + a*
x])/(448*b^2*x^(7/3)) - (143*a^3*Sqrt[b*x^(2/3) + a*x])/(4480*b^3*x^2) + (1287*a^4*Sqrt[b*x^(2/3) + a*x])/(358
40*b^4*x^(5/3)) - (429*a^5*Sqrt[b*x^(2/3) + a*x])/(10240*b^5*x^(4/3)) + (429*a^6*Sqrt[b*x^(2/3) + a*x])/(8192*
b^6*x) - (1287*a^7*Sqrt[b*x^(2/3) + a*x])/(16384*b^7*x^(2/3)) + (1287*a^8*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(
2/3) + a*x]])/(16384*b^(15/2))

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^4} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}+\frac {1}{16} a \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}-\frac {\left (13 a^2\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{224 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}+\frac {\left (143 a^3\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{2688 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}-\frac {\left (429 a^4\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{8960 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}+\frac {\left (429 a^5\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{10240 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}-\frac {\left (143 a^6\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{4096 b^5}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac {429 a^6 \sqrt {b x^{2/3}+a x}}{8192 b^6 x}+\frac {\left (429 a^7\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{16384 b^6}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac {429 a^6 \sqrt {b x^{2/3}+a x}}{8192 b^6 x}-\frac {1287 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^7 x^{2/3}}-\frac {\left (429 a^8\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac {429 a^6 \sqrt {b x^{2/3}+a x}}{8192 b^6 x}-\frac {1287 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^7 x^{2/3}}+\frac {\left (1287 a^8\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 )}{16384 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac {429 a^6 \sqrt {b x^{2/3}+a x}}{8192 b^6 x}-\frac {1287 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^7 x^{2/3}}+\frac {1287 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{16384 b^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 57, normalized size = 0.21 \begin {gather*} -\frac {2 a^8 \left (a \sqrt [3]{x}+b\right ) \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {3}{2},9;\frac {5}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^9 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^8*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[3/2, 9, 5/2, 1 + (a*x^(1/3))/b])/(b^9*x^(1/3))

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IntegrateAlgebraic [A]  time = 0.29, size = 149, normalized size = 0.56 \begin {gather*} \frac {1287 a^8 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{16384 b^{15/2}}+\frac {\sqrt {a x+b x^{2/3}} \left (-45045 a^7 x^{7/3}+30030 a^6 b x^2-24024 a^5 b^2 x^{5/3}+20592 a^4 b^3 x^{4/3}-18304 a^3 b^4 x+16640 a^2 b^5 x^{2/3}-15360 a b^6 \sqrt [3]{x}-215040 b^7\right )}{573440 b^7 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-215040*b^7 - 15360*a*b^6*x^(1/3) + 16640*a^2*b^5*x^(2/3) - 18304*a^3*b^4*x + 20592*a^
4*b^3*x^(4/3) - 24024*a^5*b^2*x^(5/3) + 30030*a^6*b*x^2 - 45045*a^7*x^(7/3)))/(573440*b^7*x^3) + (1287*a^8*Arc
Tanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(16384*b^(15/2))

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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((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.35, size = 177, normalized size = 0.67 \begin {gather*} -\frac {\frac {45045 \, a^{9} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{9} - 345345 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{9} b + 1150149 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{9} b^{2} - 2167737 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{9} b^{3} + 2518087 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{9} b^{4} - 1831739 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{9} b^{5} + 801535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{9} b^{6} + 45045 \, \sqrt {a x^{\frac {1}{3}} + b} a^{9} b^{7}}{a^{8} b^{7} x^{\frac {8}{3}}}}{573440 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="giac")

[Out]

-1/573440*(45045*a^9*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(15/2)*a^9 -
345345*(a*x^(1/3) + b)^(13/2)*a^9*b + 1150149*(a*x^(1/3) + b)^(11/2)*a^9*b^2 - 2167737*(a*x^(1/3) + b)^(9/2)*
a^9*b^3 + 2518087*(a*x^(1/3) + b)^(7/2)*a^9*b^4 - 1831739*(a*x^(1/3) + b)^(5/2)*a^9*b^5 + 801535*(a*x^(1/3) +
b)^(3/2)*a^9*b^6 + 45045*sqrt(a*x^(1/3) + b)*a^9*b^7)/(a^8*b^7*x^(8/3)))/a

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maple [A]  time = 0.06, size = 167, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {a x +b \,x^{\frac {2}{3}}}\, \left (-45045 a^{8} b^{7} x^{\frac {8}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )+45045 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {29}{2}}+801535 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} b^{\frac {27}{2}}-1831739 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {5}{2}} b^{\frac {25}{2}}+2518087 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {7}{2}} b^{\frac {23}{2}}-2167737 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {9}{2}} b^{\frac {21}{2}}+1150149 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {11}{2}} b^{\frac {19}{2}}-345345 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {13}{2}} b^{\frac {17}{2}}+45045 \left (a \,x^{\frac {1}{3}}+b \right )^{\frac {15}{2}} b^{\frac {15}{2}}\right )}{573440 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {29}{2}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b*x^(2/3))^(1/2)/x^4,x)

[Out]

-1/573440*(a*x+b*x^(2/3))^(1/2)*(45045*b^(15/2)*(a*x^(1/3)+b)^(15/2)-345345*b^(17/2)*(a*x^(1/3)+b)^(13/2)+1150
149*b^(19/2)*(a*x^(1/3)+b)^(11/2)-2167737*b^(21/2)*(a*x^(1/3)+b)^(9/2)+2518087*b^(23/2)*(a*x^(1/3)+b)^(7/2)-18
31739*b^(25/2)*(a*x^(1/3)+b)^(5/2)+801535*b^(27/2)*(a*x^(1/3)+b)^(3/2)-45045*arctanh((a*x^(1/3)+b)^(1/2)/b^(1/
2))*b^7*x^(8/3)*a^8+45045*b^(29/2)*(a*x^(1/3)+b)^(1/2))/x^3/(a*x^(1/3)+b)^(1/2)/b^(29/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^(2/3))^(1/2)/x^4,x)

[Out]

int((a*x + b*x^(2/3))^(1/2)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**(2/3)+a*x)**(1/2)/x**4,x)

[Out]

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

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