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

Optimal. Leaf size=116 \[ \frac {35 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b^2}{4 a^4 \sqrt {b x-a}}-\frac {35 b^2}{12 a^3 (b x-a)^{3/2}}+\frac {7 b}{4 a^2 x (b x-a)^{3/2}}+\frac {1}{2 a x^2 (b x-a)^{3/2}} \]

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 205} \begin {gather*} \frac {35 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 \sqrt {b x-a}}{6 a^3 x^2}+\frac {14}{3 a^2 x^2 \sqrt {b x-a}}+\frac {35 b \sqrt {b x-a}}{4 a^4 x}-\frac {2}{3 a x^2 (b x-a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(-a + b*x)^(5/2)),x]

[Out]

-2/(3*a*x^2*(-a + b*x)^(3/2)) + 14/(3*a^2*x^2*Sqrt[-a + b*x]) + (35*Sqrt[-a + b*x])/(6*a^3*x^2) + (35*b*Sqrt[-
a + b*x])/(4*a^4*x) + (35*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/(4*a^(9/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin {align*} \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}-\frac {7 \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx}{3 a}\\ &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}+\frac {14}{3 a^2 x^2 \sqrt {-a+b x}}+\frac {35 \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx}{3 a^2}\\ &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}+\frac {14}{3 a^2 x^2 \sqrt {-a+b x}}+\frac {35 \sqrt {-a+b x}}{6 a^3 x^2}+\frac {(35 b) \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{4 a^3}\\ &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}+\frac {14}{3 a^2 x^2 \sqrt {-a+b x}}+\frac {35 \sqrt {-a+b x}}{6 a^3 x^2}+\frac {35 b \sqrt {-a+b x}}{4 a^4 x}+\frac {\left (35 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^4}\\ &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}+\frac {14}{3 a^2 x^2 \sqrt {-a+b x}}+\frac {35 \sqrt {-a+b x}}{6 a^3 x^2}+\frac {35 b \sqrt {-a+b x}}{4 a^4 x}+\frac {(35 b) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^4}\\ &=-\frac {2}{3 a x^2 (-a+b x)^{3/2}}+\frac {14}{3 a^2 x^2 \sqrt {-a+b x}}+\frac {35 \sqrt {-a+b x}}{6 a^3 x^2}+\frac {35 b \sqrt {-a+b x}}{4 a^4 x}+\frac {35 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 38, normalized size = 0.33 \begin {gather*} -\frac {2 b^2 \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};1-\frac {b x}{a}\right )}{3 a^3 (b x-a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(-a + b*x)^(5/2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 93, normalized size = 0.80 \begin {gather*} \frac {35 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{9/2}}-\frac {8 a^3-56 a^2 (b x-a)-175 a (b x-a)^2-105 (b x-a)^3}{12 a^4 x^2 (b x-a)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(8*a^3 - 56*a^2*(-a + b*x) - 175*a*(-a + b*x)^2 - 105*(-a + b*x)^3)/(a^4*x^2*(-a + b*x)^(3/2)) + (35*b^2
*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/(4*a^(9/2))

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fricas [A]  time = 0.96, size = 260, normalized size = 2.24 \begin {gather*} \left [-\frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{24 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, \frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{12 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(105*(b^4*x^4 - 2*a*b^3*x^3 + a^2*b^2*x^2)*sqrt(-a)*log((b*x - 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) - 2*(
105*a*b^3*x^3 - 140*a^2*b^2*x^2 + 21*a^3*b*x + 6*a^4)*sqrt(b*x - a))/(a^5*b^2*x^4 - 2*a^6*b*x^3 + a^7*x^2), 1/
12*(105*(b^4*x^4 - 2*a*b^3*x^3 + a^2*b^2*x^2)*sqrt(a)*arctan(sqrt(b*x - a)/sqrt(a)) + (105*a*b^3*x^3 - 140*a^2
*b^2*x^2 + 21*a^3*b*x + 6*a^4)*sqrt(b*x - a))/(a^5*b^2*x^4 - 2*a^6*b*x^3 + a^7*x^2)]

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giac [A]  time = 0.90, size = 97, normalized size = 0.84 \begin {gather*} \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} + \frac {2 \, {\left (9 \, {\left (b x - a\right )} b^{2} - a b^{2}\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{4}} + \frac {11 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 13 \, \sqrt {b x - a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/4*b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(9/2) + 2/3*(9*(b*x - a)*b^2 - a*b^2)/((b*x - a)^(3/2)*a^4) + 1/4*(11
*(b*x - a)^(3/2)*b^2 + 13*sqrt(b*x - a)*a*b^2)/(a^4*b^2*x^2)

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maple [A]  time = 0.02, size = 92, normalized size = 0.79 \begin {gather*} -\frac {2 b^{2}}{3 \left (b x -a \right )^{\frac {3}{2}} a^{3}}+\frac {35 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {9}{2}}}+\frac {6 b^{2}}{\sqrt {b x -a}\, a^{4}}+\frac {13 \sqrt {b x -a}}{4 a^{3} x^{2}}+\frac {11 \left (b x -a \right )^{\frac {3}{2}}}{4 a^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*b^2/a^3/(b*x-a)^(3/2)+6*b^2/a^4/(b*x-a)^(1/2)+11/4/a^4/x^2*(b*x-a)^(3/2)+13/4/a^3/x^2*(b*x-a)^(1/2)+35/4*
b^2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(9/2)

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maxima [A]  time = 2.87, size = 121, normalized size = 1.04 \begin {gather*} \frac {105 \, {\left (b x - a\right )}^{3} b^{2} + 175 \, {\left (b x - a\right )}^{2} a b^{2} + 56 \, {\left (b x - a\right )} a^{2} b^{2} - 8 \, a^{3} b^{2}}{12 \, {\left ({\left (b x - a\right )}^{\frac {7}{2}} a^{4} + 2 \, {\left (b x - a\right )}^{\frac {5}{2}} a^{5} + {\left (b x - a\right )}^{\frac {3}{2}} a^{6}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(105*(b*x - a)^3*b^2 + 175*(b*x - a)^2*a*b^2 + 56*(b*x - a)*a^2*b^2 - 8*a^3*b^2)/((b*x - a)^(7/2)*a^4 + 2
*(b*x - a)^(5/2)*a^5 + (b*x - a)^(3/2)*a^6) + 35/4*b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(9/2)

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mupad [B]  time = 0.07, size = 117, normalized size = 1.01 \begin {gather*} \frac {35\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{9/2}}-\frac {\frac {2\,b^2}{3\,a}-\frac {175\,b^2\,{\left (a-b\,x\right )}^2}{12\,a^3}+\frac {35\,b^2\,{\left (a-b\,x\right )}^3}{4\,a^4}+\frac {14\,b^2\,\left (a-b\,x\right )}{3\,a^2}}{2\,a\,{\left (b\,x-a\right )}^{5/2}+{\left (b\,x-a\right )}^{7/2}+a^2\,{\left (b\,x-a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(b*x - a)^(5/2)),x)

[Out]

(35*b^2*atan((b*x - a)^(1/2)/a^(1/2)))/(4*a^(9/2)) - ((2*b^2)/(3*a) - (175*b^2*(a - b*x)^2)/(12*a^3) + (35*b^2
*(a - b*x)^3)/(4*a^4) + (14*b^2*(a - b*x))/(3*a^2))/(2*a*(b*x - a)^(5/2) + (b*x - a)^(7/2) + a^2*(b*x - a)^(3/
2))

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sympy [B]  time = 11.22, size = 1108, normalized size = 9.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((12*I*a**(89/2)*b**75*x**75/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**
(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) + 42*I*a**(87/2)*b**76*x**76/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/
(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) - 280*I*a**(85/2)*b**77*x**77/(24*a**(93/2)
*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) + 210*I*a**(8
3/2)*b**78*x**78/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sq
rt(a/(b*x) - 1)) + 210*I*a**42*b**(155/2)*x**(155/2)*sqrt(a/(b*x) - 1)*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a*
*(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) - 105*
pi*a**42*b**(155/2)*x**(155/2)*sqrt(a/(b*x) - 1)/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a*
*(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) - 210*I*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) - 1)*acosh(s
qrt(a)/(sqrt(b)*sqrt(x)))/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(
157/2)*sqrt(a/(b*x) - 1)) + 105*pi*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) - 1)/(24*a**(93/2)*b**(151/2)*x**(
155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)), Abs(a/(b*x)) > 1), (-6*a**(8
9/2)*b**75*x**75/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(-a/(b*x) + 1) - 12*a**(91/2)*b**(153/2)*x**(157/2)*s
qrt(-a/(b*x) + 1)) - 21*a**(87/2)*b**76*x**76/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(-a/(b*x) + 1) - 12*a**(
91/2)*b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)) + 140*a**(85/2)*b**77*x**77/(12*a**(93/2)*b**(151/2)*x**(155/2
)*sqrt(-a/(b*x) + 1) - 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)) - 105*a**(83/2)*b**78*x**78/(12*
a**(93/2)*b**(151/2)*x**(155/2)*sqrt(-a/(b*x) + 1) - 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)) -
105*a**42*b**(155/2)*x**(155/2)*sqrt(-a/(b*x) + 1)*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*a**(93/2)*b**(151/2)*x*
*(155/2)*sqrt(-a/(b*x) + 1) - 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)) + 105*a**41*b**(157/2)*x*
*(157/2)*sqrt(-a/(b*x) + 1)*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(-a/(b*x)
+ 1) - 12*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)), True))

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