3.1.87 \(\int \frac {1}{x^{7/2} (b \sqrt {x}+a x)^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac {8192 a^6 \sqrt {a x+b \sqrt {x}}}{429 b^8 \sqrt {x}}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{429 b^7 x}-\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{143 b^6 x^{3/2}}+\frac {2560 a^3 \sqrt {a x+b \sqrt {x}}}{429 b^5 x^2}-\frac {2240 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^4 x^{5/2}}+\frac {672 a \sqrt {a x+b \sqrt {x}}}{143 b^3 x^3}-\frac {56 \sqrt {a x+b \sqrt {x}}}{13 b^2 x^{7/2}}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}} \]

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Rubi [A]  time = 0.35, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2015, 2016, 2014} \begin {gather*} -\frac {1024 a^4 \sqrt {a x+b \sqrt {x}}}{143 b^6 x^{3/2}}+\frac {2560 a^3 \sqrt {a x+b \sqrt {x}}}{429 b^5 x^2}-\frac {2240 a^2 \sqrt {a x+b \sqrt {x}}}{429 b^4 x^{5/2}}-\frac {8192 a^6 \sqrt {a x+b \sqrt {x}}}{429 b^8 \sqrt {x}}+\frac {4096 a^5 \sqrt {a x+b \sqrt {x}}}{429 b^7 x}+\frac {672 a \sqrt {a x+b \sqrt {x}}}{143 b^3 x^3}-\frac {56 \sqrt {a x+b \sqrt {x}}}{13 b^2 x^{7/2}}+\frac {4}{b x^3 \sqrt {a x+b \sqrt {x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

4/(b*x^3*Sqrt[b*Sqrt[x] + a*x]) - (56*Sqrt[b*Sqrt[x] + a*x])/(13*b^2*x^(7/2)) + (672*a*Sqrt[b*Sqrt[x] + a*x])/
(143*b^3*x^3) - (2240*a^2*Sqrt[b*Sqrt[x] + a*x])/(429*b^4*x^(5/2)) + (2560*a^3*Sqrt[b*Sqrt[x] + a*x])/(429*b^5
*x^2) - (1024*a^4*Sqrt[b*Sqrt[x] + a*x])/(143*b^6*x^(3/2)) + (4096*a^5*Sqrt[b*Sqrt[x] + a*x])/(429*b^7*x) - (8
192*a^6*Sqrt[b*Sqrt[x] + a*x])/(429*b^8*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^{7/2} \left (b \sqrt {x}+a x\right )^{3/2}} \, dx &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}+\frac {14 \int \frac {1}{x^4 \sqrt {b \sqrt {x}+a x}} \, dx}{b}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}-\frac {(168 a) \int \frac {1}{x^{7/2} \sqrt {b \sqrt {x}+a x}} \, dx}{13 b^2}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}+\frac {\left (1680 a^2\right ) \int \frac {1}{x^3 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^3}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}-\frac {\left (4480 a^3\right ) \int \frac {1}{x^{5/2} \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^4}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}+\frac {\left (1280 a^4\right ) \int \frac {1}{x^2 \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^5}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}-\frac {\left (1024 a^5\right ) \int \frac {1}{x^{3/2} \sqrt {b \sqrt {x}+a x}} \, dx}{143 b^6}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}+\frac {\left (2048 a^6\right ) \int \frac {1}{x \sqrt {b \sqrt {x}+a x}} \, dx}{429 b^7}\\ &=\frac {4}{b x^3 \sqrt {b \sqrt {x}+a x}}-\frac {56 \sqrt {b \sqrt {x}+a x}}{13 b^2 x^{7/2}}+\frac {672 a \sqrt {b \sqrt {x}+a x}}{143 b^3 x^3}-\frac {2240 a^2 \sqrt {b \sqrt {x}+a x}}{429 b^4 x^{5/2}}+\frac {2560 a^3 \sqrt {b \sqrt {x}+a x}}{429 b^5 x^2}-\frac {1024 a^4 \sqrt {b \sqrt {x}+a x}}{143 b^6 x^{3/2}}+\frac {4096 a^5 \sqrt {b \sqrt {x}+a x}}{429 b^7 x}-\frac {8192 a^6 \sqrt {b \sqrt {x}+a x}}{429 b^8 \sqrt {x}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 107, normalized size = 0.48 \begin {gather*} -\frac {4 \left (2048 a^7 x^{7/2}+1024 a^6 b x^3-256 a^5 b^2 x^{5/2}+128 a^4 b^3 x^2-80 a^3 b^4 x^{3/2}+56 a^2 b^5 x-42 a b^6 \sqrt {x}+33 b^7\right )}{429 b^8 x^3 \sqrt {a x+b \sqrt {x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

(-4*(33*b^7 - 42*a*b^6*Sqrt[x] + 56*a^2*b^5*x - 80*a^3*b^4*x^(3/2) + 128*a^4*b^3*x^2 - 256*a^5*b^2*x^(5/2) + 1
024*a^6*b*x^3 + 2048*a^7*x^(7/2)))/(429*b^8*x^3*Sqrt[b*Sqrt[x] + a*x])

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IntegrateAlgebraic [A]  time = 0.36, size = 120, normalized size = 0.54 \begin {gather*} -\frac {4 \sqrt {a x+b \sqrt {x}} \left (2048 a^7 x^{7/2}+1024 a^6 b x^3-256 a^5 b^2 x^{5/2}+128 a^4 b^3 x^2-80 a^3 b^4 x^{3/2}+56 a^2 b^5 x-42 a b^6 \sqrt {x}+33 b^7\right )}{429 b^8 x^{7/2} \left (a \sqrt {x}+b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(7/2)*(b*Sqrt[x] + a*x)^(3/2)),x]

[Out]

(-4*Sqrt[b*Sqrt[x] + a*x]*(33*b^7 - 42*a*b^6*Sqrt[x] + 56*a^2*b^5*x - 80*a^3*b^4*x^(3/2) + 128*a^4*b^3*x^2 - 2
56*a^5*b^2*x^(5/2) + 1024*a^6*b*x^3 + 2048*a^7*x^(7/2)))/(429*b^8*(b + a*Sqrt[x])*x^(7/2))

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fricas [A]  time = 0.48, size = 123, normalized size = 0.55 \begin {gather*} \frac {4 \, {\left (1024 \, a^{7} b x^{4} - 384 \, a^{5} b^{3} x^{3} - 136 \, a^{3} b^{5} x^{2} - 75 \, a b^{7} x - {\left (2048 \, a^{8} x^{4} - 1280 \, a^{6} b^{2} x^{3} - 208 \, a^{4} b^{4} x^{2} - 98 \, a^{2} b^{6} x - 33 \, b^{8}\right )} \sqrt {x}\right )} \sqrt {a x + b \sqrt {x}}}{429 \, {\left (a^{2} b^{8} x^{5} - b^{10} x^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="fricas")

[Out]

4/429*(1024*a^7*b*x^4 - 384*a^5*b^3*x^3 - 136*a^3*b^5*x^2 - 75*a*b^7*x - (2048*a^8*x^4 - 1280*a^6*b^2*x^3 - 20
8*a^4*b^4*x^2 - 98*a^2*b^6*x - 33*b^8)*sqrt(x))*sqrt(a*x + b*sqrt(x))/(a^2*b^8*x^5 - b^10*x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)

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maple [C]  time = 0.09, size = 636, normalized size = 2.85 \begin {gather*} \frac {2 \sqrt {a x +b \sqrt {x}}\, \left (-1287 a^{9} b \,x^{\frac {17}{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 )+1287 a^{9} b \,x^{\frac {17}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-2574 a^{8} b^{2} x^{8} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2574 a^{8} b^{2} x^{8} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )-1287 a^{7} b^{3} x^{\frac {15}{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 )+1287 a^{7} b^{3} x^{\frac {15}{2}} \ln \left (\frac {2 a \sqrt {x}+b +2 \sqrt {a x +b \sqrt {x}}\, \sqrt {a}}{2 \sqrt {a}}\right )+2574 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {19}{2}} x^{\frac {17}{2}}+2574 \sqrt {a x +b \sqrt {x}}\, a^{\frac {19}{2}} x^{\frac {17}{2}}+5148 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {17}{2}} b \,x^{8}+5148 \sqrt {a x +b \sqrt {x}}\, a^{\frac {17}{2}} b \,x^{8}+2574 \sqrt {a x +b \sqrt {x}}\, a^{\frac {15}{2}} b^{2} x^{\frac {15}{2}}+2574 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, a^{\frac {15}{2}} b^{2} x^{\frac {15}{2}}-6006 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {17}{2}} x^{\frac {15}{2}}+858 \left (\left (a \sqrt {x}+b \right ) \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {17}{2}} x^{\frac {15}{2}}-9244 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {15}{2}} b \,x^{7}-2048 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {13}{2}} b^{2} x^{\frac {13}{2}}+512 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {11}{2}} b^{3} x^{6}-256 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {9}{2}} b^{4} x^{\frac {11}{2}}+160 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{5} x^{5}-112 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{6} x^{\frac {9}{2}}+84 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} a^{\frac {3}{2}} b^{7} x^{4}-66 \left (a x +b \sqrt {x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{8} x^{\frac {7}{2}}\right )}{429 \sqrt {\left (a \sqrt {x}+b \right ) \sqrt {x}}\, \left (a \sqrt {x}+b \right )^{2} \sqrt {a}\, b^{9} x^{\frac {15}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(a*x+b*x^(1/2))^(3/2),x)

[Out]

2/429*(a*x+b*x^(1/2))^(1/2)*(2574*x^(15/2)*(a*x+b*x^(1/2))^(1/2)*a^(15/2)*b^2+160*x^5*(a*x+b*x^(1/2))^(3/2)*a^
(7/2)*b^5-2048*x^(13/2)*(a*x+b*x^(1/2))^(3/2)*a^(13/2)*b^2+5148*x^8*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(17/2)*b-9
244*x^7*(a*x+b*x^(1/2))^(3/2)*a^(15/2)*b+5148*x^8*(a*x+b*x^(1/2))^(1/2)*a^(17/2)*b-256*x^(11/2)*(a*x+b*x^(1/2)
)^(3/2)*a^(9/2)*b^4+512*x^6*(a*x+b*x^(1/2))^(3/2)*a^(11/2)*b^3+2574*x^(15/2)*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(
15/2)*b^2+2574*x^(17/2)*((a*x^(1/2)+b)*x^(1/2))^(1/2)*a^(19/2)-66*x^(7/2)*(a*x+b*x^(1/2))^(3/2)*a^(1/2)*b^8-60
06*x^(15/2)*(a*x+b*x^(1/2))^(3/2)*a^(17/2)+2574*x^(17/2)*(a*x+b*x^(1/2))^(1/2)*a^(19/2)-1287*x^(17/2)*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^9*b+1287*x^(17/2)*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^9*b-2574*x^8*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^8*b^2+2574*x^8*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^8*b^2+858*x^
(15/2)*((a*x^(1/2)+b)*x^(1/2))^(3/2)*a^(17/2)-1287*x^(15/2)*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^3+1287*x^(15/2)*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^3-112*x^(9/2)*(a*x+b*x^(1/2))^(3/2)*a^(5/2)*b^6+84*x^4*(a*x+b*x^(1/2))^(3/2)*a^(3/2)*b^7)/((a*x^(1/2)+b)*x
^(1/2))^(1/2)/b^9/x^(15/2)/a^(1/2)/(a*x^(1/2)+b)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x + b \sqrt {x}\right )}^{\frac {3}{2}} x^{\frac {7}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^(1/2)+a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((a*x + b*sqrt(x))^(3/2)*x^(7/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^{7/2}\,{\left (a\,x+b\,\sqrt {x}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)),x)

[Out]

int(1/(x^(7/2)*(a*x + b*x^(1/2))^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {7}{2}} \left (a x + b \sqrt {x}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(b*x**(1/2)+a*x)**(3/2),x)

[Out]

Integral(1/(x**(7/2)*(a*x + b*sqrt(x))**(3/2)), x)

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