3.539 \(\int \frac{x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

(2*(A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 5*a*B)*x^(3/2))/(3*a*b^2*Sqrt[a + b*x]) - ((2*A*
b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(a*b^3) + ((2*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

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Rubi [A]  time = 0.0521983, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {78, 47, 50, 63, 217, 206} \[ \frac{2 x^{3/2} (2 A b-5 a B)}{3 a b^2 \sqrt{a+b x}}-\frac{\sqrt{x} \sqrt{a+b x} (2 A b-5 a B)}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{2 x^{5/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 5*a*B)*x^(3/2))/(3*a*b^2*Sqrt[a + b*x]) - ((2*A*
b - 5*a*B)*Sqrt[x]*Sqrt[a + b*x])/(a*b^3) + ((2*A*b - 5*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

Rule 78

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

Rule 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{x^{3/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}-\frac{\left (2 \left (A b-\frac{5 a B}{2}\right )\right ) \int \frac{x^{3/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \int \frac{\sqrt{x}}{\sqrt{a+b x}} \, dx}{a b^2}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx}{2 b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{a+b x}}\right )}{b^3}\\ &=\frac{2 (A b-a B) x^{5/2}}{3 a b (a+b x)^{3/2}}+\frac{2 (2 A b-5 a B) x^{3/2}}{3 a b^2 \sqrt{a+b x}}-\frac{(2 A b-5 a B) \sqrt{x} \sqrt{a+b x}}{a b^3}+\frac{(2 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.053889, size = 80, normalized size = 0.6 \[ \frac{2 x^{5/2} \left ((a+b x) \sqrt{\frac{b x}{a}+1} (5 a B-2 A b) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{7}{2};-\frac{b x}{a}\right )+5 a (A b-a B)\right )}{15 a^2 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(5*a*(A*b - a*B) + (-2*A*b + 5*a*B)*(a + b*x)*Sqrt[1 + (b*x)/a]*Hypergeometric2F1[3/2, 5/2, 7/2, -(
(b*x)/a)]))/(15*a^2*b*(a + b*x)^(3/2))

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Maple [B]  time = 0.013, size = 315, normalized size = 2.4 \begin{align*}{\frac{1}{6} \left ( 6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{b}^{3}-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}a{b}^{2}+6\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+12\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) xa{b}^{2}-16\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x-30\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x{a}^{2}b+40\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}xa+6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}b-12\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}a-15\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{3}+30\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{2} \right ) \sqrt{x}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{b}^{-{\frac{7}{2}}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*x^2*b^3-15*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2
)+2*b*x+a)/b^(1/2))*x^2*a*b^2+6*B*x^2*b^(5/2)*(x*(b*x+a))^(1/2)+12*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x
+a)/b^(1/2))*x*a*b^2-16*A*(x*(b*x+a))^(1/2)*b^(5/2)*x-30*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2
))*x*a^2*b+40*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a+6*A*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^2*b-
12*A*(x*(b*x+a))^(1/2)*b^(3/2)*a-15*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^3+30*B*(x*(b*x+a
))^(1/2)*b^(1/2)*a^2)*x^(1/2)/(x*(b*x+a))^(1/2)/b^(7/2)/(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(x^(3/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.83938, size = 729, normalized size = 5.48 \begin{align*} \left [-\frac{3 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b +{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt{b} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \,{\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{6 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{3 \,{\left (5 \, B a^{3} - 2 \, A a^{2} b +{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (3 \, B b^{3} x^{2} + 15 \, B a^{2} b - 6 \, A a b^{2} + 4 \,{\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*(5*B*a^3 - 2*A*a^2*b + (5*B*a*b^2 - 2*A*b^3)*x^2 + 2*(5*B*a^2*b - 2*A*a*b^2)*x)*sqrt(b)*log(2*b*x + 2
*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(3*B*b^3*x^2 + 15*B*a^2*b - 6*A*a*b^2 + 4*(5*B*a*b^2 - 2*A*b^3)*x)*sqr
t(b*x + a)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/3*(3*(5*B*a^3 - 2*A*a^2*b + (5*B*a*b^2 - 2*A*b^3)*x^2 +
2*(5*B*a^2*b - 2*A*a*b^2)*x)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (3*B*b^3*x^2 + 15*B*a^2*b
- 6*A*a*b^2 + 4*(5*B*a*b^2 - 2*A*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 85.9072, size = 417, normalized size = 3.14 \begin{align*} \frac{\sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a} B{\left | b \right |}}{b^{5}} + \frac{{\left (5 \, B a \sqrt{b}{\left | b \right |} - 2 \, A b^{\frac{3}{2}}{\left | b \right |}\right )} \log \left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{2 \, b^{5}} + \frac{4 \,{\left (9 \, B a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b}{\left | b \right |} + 12 \, B a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}}{\left | b \right |} - 6 \, A a{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac{3}{2}}{\left | b \right |} + 7 \, B a^{4} b^{\frac{5}{2}}{\left | b \right |} - 6 \, A a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{5}{2}}{\left | b \right |} - 4 \, A a^{3} b^{\frac{7}{2}}{\left | b \right |}\right )}}{3 \,{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*B*abs(b)/b^5 + 1/2*(5*B*a*sqrt(b)*abs(b) - 2*A*b^(3/2)*abs(b))*log((sqrt
(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^5 + 4/3*(9*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b -
a*b))^4*sqrt(b)*abs(b) + 12*B*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(3/2)*abs(b) - 6*A*a*
(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 7*B*a^4*b^(5/2)*abs(b) - 6*A*a^2*(sqrt(b*
x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(5/2)*abs(b) - 4*A*a^3*b^(7/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b)
- sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^4)