3.129 \(\int \frac {1}{x^{5/2} (b \sqrt {x}+a x)^{3/2}} \, dx\)

Optimal. Leaf size=165 \[ -\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{63 b^6 \sqrt {x}}+\frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{63 b^5 x}-\frac {128 a^2 \sqrt {a x+b \sqrt {x}}}{21 b^4 x^{3/2}}+\frac {320 a \sqrt {a x+b \sqrt {x}}}{63 b^3 x^2}-\frac {40 \sqrt {a x+b \sqrt {x}}}{9 b^2 x^{5/2}}+\frac {4}{b x^2 \sqrt {a x+b \sqrt {x}}} \]

[Out]

4/b/x^2/(b*x^(1/2)+a*x)^(1/2)-40/9*(b*x^(1/2)+a*x)^(1/2)/b^2/x^(5/2)+320/63*a*(b*x^(1/2)+a*x)^(1/2)/b^3/x^2-12
8/21*a^2*(b*x^(1/2)+a*x)^(1/2)/b^4/x^(3/2)+512/63*a^3*(b*x^(1/2)+a*x)^(1/2)/b^5/x-1024/63*a^4*(b*x^(1/2)+a*x)^
(1/2)/b^6/x^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \[ -\frac {128 a^2 \sqrt {a x+b \sqrt {x}}}{21 b^4 x^{3/2}}-\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{63 b^6 \sqrt {x}}+\frac {512 a^3 \sqrt {a x+b \sqrt {x}}}{63 b^5 x}+\frac {320 a \sqrt {a x+b \sqrt {x}}}{63 b^3 x^2}-\frac {40 \sqrt {a x+b \sqrt {x}}}{9 b^2 x^{5/2}}+\frac {4}{b x^2 \sqrt {a x+b \sqrt {x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(b*x^2*Sqrt[b*Sqrt[x] + a*x]) - (40*Sqrt[b*Sqrt[x] + a*x])/(9*b^2*x^(5/2)) + (320*a*Sqrt[b*Sqrt[x] + a*x])/(
63*b^3*x^2) - (128*a^2*Sqrt[b*Sqrt[x] + a*x])/(21*b^4*x^(3/2)) + (512*a^3*Sqrt[b*Sqrt[x] + a*x])/(63*b^5*x) -
(1024*a^4*Sqrt[b*Sqrt[x] + a*x])/(63*b^6*Sqrt[x])

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 2015

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] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n}, x] &&  !IntegerQ[p] && NeQ[n, j
] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && LtQ[p, -1] && (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 {1}{x^{5/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}+\frac {10 \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}-\frac {(80 a) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{9 b^2}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}+\frac {\left (160 a^2\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{21 b^3}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}-\frac {\left (128 a^3\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{21 b^4}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{63 b^5 x}+\frac {\left (256 a^4\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{63 b^5}\\ &=\frac {4}{b x^2 \sqrt {b \sqrt {x}+a x}}-\frac {40 \sqrt {b \sqrt {x}+a x}}{9 b^2 x^{5/2}}+\frac {320 a \sqrt {b \sqrt {x}+a x}}{63 b^3 x^2}-\frac {128 a^2 \sqrt {b \sqrt {x}+a x}}{21 b^4 x^{3/2}}+\frac {512 a^3 \sqrt {b \sqrt {x}+a x}}{63 b^5 x}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{63 b^6 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 83, normalized size = 0.50 \[ -\frac {4 \left (256 a^5 x^{5/2}+128 a^4 b x^2-32 a^3 b^2 x^{3/2}+16 a^2 b^3 x-10 a b^4 \sqrt {x}+7 b^5\right )}{63 b^6 x^2 \sqrt {a x+b \sqrt {x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(7*b^5 - 10*a*b^4*Sqrt[x] + 16*a^2*b^3*x - 32*a^3*b^2*x^(3/2) + 128*a^4*b*x^2 + 256*a^5*x^(5/2)))/(63*b^6*
x^2*Sqrt[b*Sqrt[x] + a*x])

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fricas [A]  time = 1.56, size = 101, normalized size = 0.61 \[ \frac {4 \, {\left (128 \, a^{5} b x^{3} - 48 \, a^{3} b^{3} x^{2} - 17 \, a b^{5} x - {\left (256 \, a^{6} x^{3} - 160 \, a^{4} b^{2} x^{2} - 26 \, a^{2} b^{4} x - 7 \, b^{6}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{63 \, {\left (a^{2} b^{6} x^{4} - b^{8} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/63*(128*a^5*b*x^3 - 48*a^3*b^3*x^2 - 17*a*b^5*x - (256*a^6*x^3 - 160*a^4*b^2*x^2 - 26*a^2*b^4*x - 7*b^6)*sqr
t(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^6*x^4 - b^8*x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 592, normalized size = 3.59 \[ \frac {4 \sqrt {a x +b \sqrt {x}}\, \left (-63 a^{7} b \,x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{7} b \,x^{\frac {13}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-126 a^{6} b^{2} x^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+126 a^{6} b^{2} x^{6} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-63 a^{5} b^{3} x^{\frac {11}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{5} b^{3} x^{\frac {11}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+126 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {13}{2}}+126 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} x^{\frac {13}{2}}+252 \sqrt {a x +b \sqrt {x}}\, a^{\frac {13}{2}} b \,x^{6}+252 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {13}{2}} b \,x^{6}+126 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {11}{2}} b^{2} x^{\frac {11}{2}}+126 \sqrt {a x +b \sqrt {x}}\, a^{\frac {11}{2}} b^{2} x^{\frac {11}{2}}+63 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{\frac {11}{2}}-315 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} x^{\frac {11}{2}}-508 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b \,x^{5}-128 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{2} x^{\frac {9}{2}}+32 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{3} x^{4}-16 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{4} x^{\frac {7}{2}}+10 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{5} x^{3}-7 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{6} x^{\frac {5}{2}}\right )}{63 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{7} x^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/63*(a*x+b*x^(1/2))^(1/2)*(126*x^6*ln(1/2*(2*a*x^(1/2)+b+2*(a*x+b*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*a^6*b^2+63
*x^(11/2)*((a*x^(1/2)+b)*x^(1/2))^(3/2)*a^(13/2)-16*x^(7/2)*(a*x+b*x^(1/2))^(3/2)*a^(5/2)*b^4-63*x^(11/2)*ln(1
/2*(2*a*x^(1/2)+b+2*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*a^5*b^3+63*x^(11/2)*ln(1/2*(2*a*x^(1/2)+b+
2*(a*x+b*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*a^5*b^3+10*x^3*(a*x+b*x^(1/2))^(3/2)*a^(3/2)*b^5+126*x^(11/2)*((a*x^
(1/2)+b)*x^(1/2))^(1/2)*a^(11/2)*b^2+126*x^(11/2)*(a*x+b*x^(1/2))^(1/2)*a^(11/2)*b^2+252*x^6*(a*x+b*x^(1/2))^(
1/2)*a^(13/2)*b+32*x^4*(a*x+b*x^(1/2))^(3/2)*a^(7/2)*b^3-508*x^5*(a*x+b*x^(1/2))^(3/2)*a^(11/2)*b-128*x^(9/2)*
(a*x+b*x^(1/2))^(3/2)*a^(9/2)*b^2+252*x^6*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(13/2)*b-7*x^(5/2)*(a*x+b*x^(1/2))^(
3/2)*a^(1/2)*b^6+126*x^(13/2)*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(15/2)-315*x^(11/2)*(a*x+b*x^(1/2))^(3/2)*a^(13/
2)+126*x^(13/2)*(a*x+b*x^(1/2))^(1/2)*a^(15/2)-63*x^(13/2)*ln(1/2*(2*a*x^(1/2)+b+2*((a*x^(1/2)+b)*x^(1/2))^(1/
2)*a^(1/2))/a^(1/2))*a^7*b+63*x^(13/2)*ln(1/2*(2*a*x^(1/2)+b+2*(a*x+b*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*a^7*b-1
26*x^6*ln(1/2*(2*a*x^(1/2)+b+2*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(1/2))/a^(1/2))*a^6*b^2)/((a*x^(1/2)+b)*x^(1/2)
)^(1/2)/b^7/x^(11/2)/a^(1/2)/(a*x^(1/2)+b)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{5/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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