3.2.74 \(\int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx\) [174]

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

[Out]

1287/16384*a^8*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(15/2)-3/8*(b*x^(2/3)+a*x)^(1/2)/x^3-3/112*a*(
b*x^(2/3)+a*x)^(1/2)/b/x^(8/3)+13/448*a^2*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3)-143/4480*a^3*(b*x^(2/3)+a*x)^(1/2)
/b^3/x^2+1287/35840*a^4*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(5/3)-429/10240*a^5*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+429/
8192*a^6*(b*x^(2/3)+a*x)^(1/2)/b^6/x-1287/16384*a^7*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(2/3)

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Rubi [A]
time = 0.32, 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 = {2045, 2050, 2054, 212} \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 {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} \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 212

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

Rule 2045

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 2050

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 2054

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 ) \text {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 [A]
time = 0.26, size = 149, normalized size = 0.56 \begin {gather*} \frac {\sqrt {b x^{2/3}+a x} \left (-215040 b^7-15360 a b^6 \sqrt [3]{x}+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}\right )}{573440 b^7 x^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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[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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Maple [A]
time = 0.35, size = 167, normalized size = 0.63

method	result	size
derivativedivides	\(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (45045 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {15}{2}}-345345 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {17}{2}}+1150149 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {19}{2}}-2167737 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {21}{2}}+2518087 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {23}{2}}-1831739 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {25}{2}}+801535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {27}{2}}-45045 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{8} x^{\frac {8}{3}}+45045 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {29}{2}}\right )}{573440 x^{3} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {29}{2}}}\)	\(167\)
default	\(\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-45045 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {15}{2}}+345345 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {17}{2}}-1150149 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {19}{2}}+2167737 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {21}{2}}-2518087 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {23}{2}}+1831739 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {25}{2}}+45045 \arctanh \left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{8} x^{\frac {8}{3}}-801535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {27}{2}}-45045 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {29}{2}}\right )}{573440 x^{3} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {29}{2}}}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((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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Fricas [F(-1)] Timed out
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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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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Giac [A]
time = 1.64, 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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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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