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

Optimal. Leaf size=106 \[ \frac{35 b^2}{4 a^4 \sqrt{a+b x}}+\frac{35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{7 b}{4 a^2 x (a+b x)^{3/2}}-\frac{1}{2 a x^2 (a+b x)^{3/2}} \]

[Out]

(35*b^2)/(12*a^3*(a + b*x)^(3/2)) - 1/(2*a*x^2*(a + b*x)^(3/2)) + (7*b)/(4*a^2*x*(a + b*x)^(3/2)) + (35*b^2)/(
4*a^4*Sqrt[a + b*x]) - (35*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(9/2))

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Rubi [A]  time = 0.0339514, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{35 \sqrt{a+b x}}{6 a^3 x^2}+\frac{14}{3 a^2 x^2 \sqrt{a+b x}}+\frac{35 b \sqrt{a+b x}}{4 a^4 x}+\frac{2}{3 a x^2 (a+b x)^{3/2}} \]

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*ArcTanh[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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0067251, size = 35, normalized size = 0.33 \[ \frac{2 b^2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b x}{a}+1\right )}{3 a^3 (a+b x)^{3/2}} \]

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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Maple [A]  time = 0.013, size = 80, normalized size = 0.8 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{a}^{4}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ({\frac{11\, \left ( bx+a \right ) ^{3/2}}{8}}-{\frac{13\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{35}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }+3\,{\frac{1}{{a}^{4}\sqrt{bx+a}}}+1/3\,{\frac{1}{{a}^{3} \left ( bx+a \right ) ^{3/2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2*(1/a^4*((11/8*(b*x+a)^(3/2)-13/8*a*(b*x+a)^(1/2))/b^2/x^2-35/8*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2))+3
/a^4/(b*x+a)^(1/2)+1/3/a^3/(b*x+a)^(3/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59303, size = 566, normalized size = 5.34 \begin{align*} \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}}{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{align*}

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)/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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Sympy [B]  time = 12.1094, size = 464, normalized size = 4.38 \begin{align*} - \frac{6 a^{\frac{89}{2}} b^{75} x^{75}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{21 a^{\frac{87}{2}} b^{76} x^{76}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{140 a^{\frac{85}{2}} b^{77} x^{77}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{105 a^{\frac{83}{2}} b^{78} x^{78}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{105 a^{42} b^{\frac{155}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{105 a^{41} b^{\frac{157}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{155}{2}} \sqrt{\frac{a}{b x} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{157}{2}} \sqrt{\frac{a}{b x} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a**(89/2)*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**(1
57/2)*sqrt(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**(1
55/2)*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/(1
2*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)*asinh(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)*asinh(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))

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Giac [A]  time = 1.15996, size = 126, normalized size = 1.19 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{4}} + \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{align*}

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))/(sqrt(-a)*a^4) + 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)