3.1763 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=197 \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]

[Out]

(-3*e*(4*b*B*d - 5*A*b*e + a*B*e))/(4*b*(b*d - a*e)^3*Sqrt[d + e*x]) - (A*b - a*B)/(2*b*(b*d - a*e)*(a + b*x)^
2*Sqrt[d + e*x]) - (4*b*B*d - 5*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]) + (3*e*(4*b*B*d - 5
*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d - a*e)^(7/2))

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Rubi [A]  time = 0.169148, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \[ -\frac{A b-a B}{2 b (a+b x)^2 \sqrt{d+e x} (b d-a e)}-\frac{3 e (a B e-5 A b e+4 b B d)}{4 b \sqrt{d+e x} (b d-a e)^3}-\frac{a B e-5 A b e+4 b B d}{4 b (a+b x) \sqrt{d+e x} (b d-a e)^2}+\frac{3 e (a B e-5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*e*(4*b*B*d - 5*A*b*e + a*B*e))/(4*b*(b*d - a*e)^3*Sqrt[d + e*x]) - (A*b - a*B)/(2*b*(b*d - a*e)*(a + b*x)^
2*Sqrt[d + e*x]) - (4*b*B*d - 5*A*b*e + a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*Sqrt[d + e*x]) + (3*e*(4*b*B*d - 5
*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*Sqrt[b]*(b*d - a*e)^(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 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{A+B x}{(a+b x)^3 (d+e x)^{3/2}} \, dx &=-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt{d+e x}}+\frac{(4 b B d-5 A b e+a B e) \int \frac{1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{4 b (b d-a e)}\\ &=-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt{d+e x}}-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt{d+e x}}-\frac{(3 e (4 b B d-5 A b e+a B e)) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac{3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt{d+e x}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt{d+e x}}-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt{d+e x}}-\frac{(3 e (4 b B d-5 A b e+a B e)) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 (b d-a e)^3}\\ &=-\frac{3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt{d+e x}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt{d+e x}}-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt{d+e x}}-\frac{(3 (4 b B d-5 A b e+a B e)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 (b d-a e)^3}\\ &=-\frac{3 e (4 b B d-5 A b e+a B e)}{4 b (b d-a e)^3 \sqrt{d+e x}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 \sqrt{d+e x}}-\frac{4 b B d-5 A b e+a B e}{4 b (b d-a e)^2 (a+b x) \sqrt{d+e x}}+\frac{3 e (4 b B d-5 A b e+a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{b} (b d-a e)^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0615797, size = 96, normalized size = 0.49 \[ \frac{\frac{e (-a B e+5 A b e-4 b B d) \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac{a B-A b}{(a+b x)^2}}{2 b \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-(A*b) + a*B)/(a + b*x)^2 + (e*(-4*b*B*d + 5*A*b*e - a*B*e)*Hypergeometric2F1[-1/2, 2, 1/2, (b*(d + e*x))/(b
*d - a*e)])/(b*d - a*e)^2)/(2*b*(b*d - a*e)*Sqrt[d + e*x])

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Maple [B]  time = 0.02, size = 485, normalized size = 2.5 \begin{align*} -2\,{\frac{A{e}^{2}}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+2\,{\frac{eBd}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,A{b}^{2}{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}eBd}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,Aba{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,A{b}^{2}d{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,B{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{Bbad{e}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{{b}^{2}eB{d}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,Ab{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{3\,Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+3\,{\frac{bBde}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(a*e-b*d)^3/(e*x+d)^(1/2)*A*e^2+2*e/(a*e-b*d)^3/(e*x+d)^(1/2)*B*d-7/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(3/
2)*A*b^2*e^2+3/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*b*e^2+e/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*b
^2*B*d-9/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a*b*e^3+9/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*b^2
*d*e^2+5/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*e^3-1/4/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*b
*d*e^2-e/(a*e-b*d)^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*b^2*d^2-15/4/(a*e-b*d)^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+
d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*b*e^2+3/4/(a*e-b*d)^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)
^(1/2))*B*a*e^2+3*e/(a*e-b*d)^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*b*d

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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((B*x+A)/(b*x+a)^3/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.63192, size = 2884, normalized size = 14.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*(4*B*a^2*b*d^2*e + (B*a^3 - 5*A*a^2*b)*d*e^2 + (4*B*b^3*d*e^2 + (B*a*b^2 - 5*A*b^3)*e^3)*x^3 + (4*B*b^
3*d^2*e + (9*B*a*b^2 - 5*A*b^3)*d*e^2 + 2*(B*a^2*b - 5*A*a*b^2)*e^3)*x^2 + (8*B*a*b^2*d^2*e + 2*(3*B*a^2*b - 5
*A*a*b^2)*d*e^2 + (B*a^3 - 5*A*a^2*b)*e^3)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e + 2*sqrt(b^2*d - a*
b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(8*A*a^3*b*e^3 + 2*(B*a*b^3 + A*b^4)*d^3 + 11*(B*a^2*b^2 - A*a*b^3)*d^2*e -
(13*B*a^3*b - A*a^2*b^2)*d*e^2 + 3*(4*B*b^4*d^2*e - (3*B*a*b^3 + 5*A*b^4)*d*e^2 - (B*a^2*b^2 - 5*A*a*b^3)*e^3
)*x^2 + (4*B*b^4*d^3 + (17*B*a*b^3 - 5*A*b^4)*d^2*e - 4*(4*B*a^2*b^2 + 5*A*a*b^3)*d*e^2 - 5*(B*a^3*b - 5*A*a^2
*b^2)*e^3)*x)*sqrt(e*x + d))/(a^2*b^5*d^5 - 4*a^3*b^4*d^4*e + 6*a^4*b^3*d^3*e^2 - 4*a^5*b^2*d^2*e^3 + a^6*b*d*
e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^3 + (b^7*d^5 - 2*a*b
^6*d^4*e - 2*a^2*b^5*d^3*e^2 + 8*a^3*b^4*d^2*e^3 - 7*a^4*b^3*d*e^4 + 2*a^5*b^2*e^5)*x^2 + (2*a*b^6*d^5 - 7*a^2
*b^5*d^4*e + 8*a^3*b^4*d^3*e^2 - 2*a^4*b^3*d^2*e^3 - 2*a^5*b^2*d*e^4 + a^6*b*e^5)*x), -1/4*(3*(4*B*a^2*b*d^2*e
+ (B*a^3 - 5*A*a^2*b)*d*e^2 + (4*B*b^3*d*e^2 + (B*a*b^2 - 5*A*b^3)*e^3)*x^3 + (4*B*b^3*d^2*e + (9*B*a*b^2 - 5
*A*b^3)*d*e^2 + 2*(B*a^2*b - 5*A*a*b^2)*e^3)*x^2 + (8*B*a*b^2*d^2*e + 2*(3*B*a^2*b - 5*A*a*b^2)*d*e^2 + (B*a^3
- 5*A*a^2*b)*e^3)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (8*A*a^3
*b*e^3 + 2*(B*a*b^3 + A*b^4)*d^3 + 11*(B*a^2*b^2 - A*a*b^3)*d^2*e - (13*B*a^3*b - A*a^2*b^2)*d*e^2 + 3*(4*B*b^
4*d^2*e - (3*B*a*b^3 + 5*A*b^4)*d*e^2 - (B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + (4*B*b^4*d^3 + (17*B*a*b^3 - 5*A*b^
4)*d^2*e - 4*(4*B*a^2*b^2 + 5*A*a*b^3)*d*e^2 - 5*(B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(a^2*b^5*d^5 -
4*a^3*b^4*d^4*e + 6*a^4*b^3*d^3*e^2 - 4*a^5*b^2*d^2*e^3 + a^6*b*d*e^4 + (b^7*d^4*e - 4*a*b^6*d^3*e^2 + 6*a^2*
b^5*d^2*e^3 - 4*a^3*b^4*d*e^4 + a^4*b^3*e^5)*x^3 + (b^7*d^5 - 2*a*b^6*d^4*e - 2*a^2*b^5*d^3*e^2 + 8*a^3*b^4*d^
2*e^3 - 7*a^4*b^3*d*e^4 + 2*a^5*b^2*e^5)*x^2 + (2*a*b^6*d^5 - 7*a^2*b^5*d^4*e + 8*a^3*b^4*d^3*e^2 - 2*a^4*b^3*
d^2*e^3 - 2*a^5*b^2*d*e^4 + a^6*b*e^5)*x)]

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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((B*x+A)/(b*x+a)**3/(e*x+d)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.78832, size = 467, normalized size = 2.37 \begin{align*} -\frac{3 \,{\left (4 \, B b d e + B a e^{2} - 5 \, A b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (B d e - A e^{2}\right )}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{x e + d}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} d e - 4 \, \sqrt{x e + d} B b^{2} d^{2} e + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b e^{2} - 7 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} e^{2} - \sqrt{x e + d} B a b d e^{2} + 9 \, \sqrt{x e + d} A b^{2} d e^{2} + 5 \, \sqrt{x e + d} B a^{2} e^{3} - 9 \, \sqrt{x e + d} A a b e^{3}}{4 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(4*B*b*d*e + B*a*e^2 - 5*A*b*e^2)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3 - 3*a*b^2*d^2*e
+ 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) - 2*(B*d*e - A*e^2)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2
- a^3*e^3)*sqrt(x*e + d)) - 1/4*(4*(x*e + d)^(3/2)*B*b^2*d*e - 4*sqrt(x*e + d)*B*b^2*d^2*e + 3*(x*e + d)^(3/2
)*B*a*b*e^2 - 7*(x*e + d)^(3/2)*A*b^2*e^2 - sqrt(x*e + d)*B*a*b*d*e^2 + 9*sqrt(x*e + d)*A*b^2*d*e^2 + 5*sqrt(x
*e + d)*B*a^2*e^3 - 9*sqrt(x*e + d)*A*a*b*e^3)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((x*e + d)
*b - b*d + a*e)^2)