3.1764 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=240 \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

[Out]

(-5*e*(4*b*B*d - 7*A*b*e + 3*a*B*e))/(12*b*(b*d - a*e)^3*(d + e*x)^(3/2)) - (A*b - a*B)/(2*b*(b*d - a*e)*(a +
b*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 7*A*b*e + 3*a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)) - (5*e*(
4*b*B*d - 7*A*b*e + 3*a*B*e))/(4*(b*d - a*e)^4*Sqrt[d + e*x]) + (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*a*B*e)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

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Rubi [A]  time = 0.210945, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 208} \[ -\frac{A b-a B}{2 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{4 \sqrt{d+e x} (b d-a e)^4}-\frac{5 e (3 a B e-7 A b e+4 b B d)}{12 b (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-7 A b e+4 b B d}{4 b (a+b x) (d+e x)^{3/2} (b d-a e)^2}+\frac{5 \sqrt{b} e (3 a B e-7 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*e*(4*b*B*d - 7*A*b*e + 3*a*B*e))/(12*b*(b*d - a*e)^3*(d + e*x)^(3/2)) - (A*b - a*B)/(2*b*(b*d - a*e)*(a +
b*x)^2*(d + e*x)^(3/2)) - (4*b*B*d - 7*A*b*e + 3*a*B*e)/(4*b*(b*d - a*e)^2*(a + b*x)*(d + e*x)^(3/2)) - (5*e*(
4*b*B*d - 7*A*b*e + 3*a*B*e))/(4*(b*d - a*e)^4*Sqrt[d + e*x]) + (5*Sqrt[b]*e*(4*b*B*d - 7*A*b*e + 3*a*B*e)*Arc
Tanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b*d - a*e)^(9/2))

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)^{5/2}} \, dx &=-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac{(4 b B d-7 A b e+3 a B e) \int \frac{1}{(a+b x)^2 (d+e x)^{5/2}} \, dx}{4 b (b d-a e)}\\ &=-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac{(5 e (4 b B d-7 A b e+3 a B e)) \int \frac{1}{(a+b x) (d+e x)^{5/2}} \, dx}{8 b (b d-a e)^2}\\ &=-\frac{5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac{(5 e (4 b B d-7 A b e+3 a B e)) \int \frac{1}{(a+b x) (d+e x)^{3/2}} \, dx}{8 (b d-a e)^3}\\ &=-\frac{5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac{5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt{d+e x}}-\frac{(5 b e (4 b B d-7 A b e+3 a B e)) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 (b d-a e)^4}\\ &=-\frac{5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac{5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt{d+e x}}-\frac{(5 b (4 b B d-7 A b e+3 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)^4}\\ &=-\frac{5 e (4 b B d-7 A b e+3 a B e)}{12 b (b d-a e)^3 (d+e x)^{3/2}}-\frac{A b-a B}{2 b (b d-a e) (a+b x)^2 (d+e x)^{3/2}}-\frac{4 b B d-7 A b e+3 a B e}{4 b (b d-a e)^2 (a+b x) (d+e x)^{3/2}}-\frac{5 e (4 b B d-7 A b e+3 a B e)}{4 (b d-a e)^4 \sqrt{d+e x}}+\frac{5 \sqrt{b} e (4 b B d-7 A b e+3 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 (b d-a e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0609988, size = 97, normalized size = 0.4 \[ \frac{\frac{e (-3 a B e+7 A b e-4 b B d) \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac{3 (a B-A b)}{(a+b x)^2}}{6 b (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(-(A*b) + a*B))/(a + b*x)^2 + (e*(-4*b*B*d + 7*A*b*e - 3*a*B*e)*Hypergeometric2F1[-3/2, 2, -1/2, (b*(d + e
*x))/(b*d - a*e)])/(b*d - a*e)^2)/(6*b*(b*d - a*e)*(d + e*x)^(3/2))

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Maple [B]  time = 0.023, size = 568, normalized size = 2.4 \begin{align*} -{\frac{2\,A{e}^{2}}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,eBd}{3\, \left ( ae-bd \right ) ^{3}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+6\,{\frac{Ab{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-2\,{\frac{Ba{e}^{2}}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}-4\,{\frac{bBde}{ \left ( ae-bd \right ) ^{4}\sqrt{ex+d}}}+{\frac{11\,{b}^{3}A{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{b}^{2}Ba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{3}Bde}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{13\,a{b}^{2}A{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{13\,{b}^{3}Ad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{9\,bB{a}^{2}{e}^{3}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{5\,{b}^{2}Bad{e}^{2}}{4\, \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{e{b}^{3}B{d}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bxe+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{35\,A{b}^{2}{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-{\frac{15\,Bba{e}^{2}}{4\, \left ( ae-bd \right ) ^{4}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-5\,{\frac{{b}^{2}eBd}{ \left ( ae-bd \right ) ^{4}\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)^(5/2),x)

[Out]

-2/3/(a*e-b*d)^3/(e*x+d)^(3/2)*A*e^2+2/3*e/(a*e-b*d)^3/(e*x+d)^(3/2)*B*d+6/(a*e-b*d)^4/(e*x+d)^(1/2)*A*b*e^2-2
/(a*e-b*d)^4/(e*x+d)^(1/2)*B*a*e^2-4*e/(a*e-b*d)^4/(e*x+d)^(1/2)*B*b*d+11/4/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x
+d)^(3/2)*A*e^2-7/4/(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a*e^2-e/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d
)^(3/2)*B*d+13/4/(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a*e^3-13/4/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d
)^(1/2)*A*d*e^2-9/4/(a*e-b*d)^4*b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*e^3+5/4/(a*e-b*d)^4*b^2/(b*e*x+a*e)^2*(e*x
+d)^(1/2)*B*a*d*e^2+e/(a*e-b*d)^4*b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*d^2+35/4/(a*e-b*d)^4*b^2/((a*e-b*d)*b)^(1/
2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*e^2-15/4/(a*e-b*d)^4*b/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(
1/2)/((a*e-b*d)*b)^(1/2))*B*a*e^2-5*e/(a*e-b*d)^4*b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)
^(1/2))*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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.60677, size = 3665, normalized size = 15.27 \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)^(5/2),x, algorithm="fricas")

[Out]

[-1/24*(15*(4*B*a^2*b*d^3*e + (3*B*a^3 - 7*A*a^2*b)*d^2*e^2 + (4*B*b^3*d*e^3 + (3*B*a*b^2 - 7*A*b^3)*e^4)*x^4
+ 2*(4*B*b^3*d^2*e^2 + 7*(B*a*b^2 - A*b^3)*d*e^3 + (3*B*a^2*b - 7*A*a*b^2)*e^4)*x^3 + (4*B*b^3*d^3*e + (19*B*a
*b^2 - 7*A*b^3)*d^2*e^2 + 4*(4*B*a^2*b - 7*A*a*b^2)*d*e^3 + (3*B*a^3 - 7*A*a^2*b)*e^4)*x^2 + 2*(4*B*a*b^2*d^3*
e + 7*(B*a^2*b - A*a*b^2)*d^2*e^2 + (3*B*a^3 - 7*A*a^2*b)*d*e^3)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a
*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(8*A*a^3*e^3 + 6*(B*a*b^2 + A*b^3)*d^3 +
(83*B*a^2*b - 39*A*a*b^2)*d^2*e + 16*(B*a^3 - 5*A*a^2*b)*d*e^2 + 15*(4*B*b^3*d*e^2 + (3*B*a*b^2 - 7*A*b^3)*e^3
)*x^3 + 5*(16*B*b^3*d^2*e + 4*(8*B*a*b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*A*a*b^2)*e^3)*x^2 + (12*B*b^3*d^3
+ (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(67*B*a^2*b - 119*A*a*b^2)*d*e^2 + 8*(3*B*a^3 - 7*A*a^2*b)*e^3)*x)*sqrt(
e*x + d))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 -
4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*
a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3
*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b
^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x), 1/12*(15*(4*B*a^2*b*d^3*e + (3*B*a^3 - 7*A*a^2*b)*d^2*e^2 + (4*B
*b^3*d*e^3 + (3*B*a*b^2 - 7*A*b^3)*e^4)*x^4 + 2*(4*B*b^3*d^2*e^2 + 7*(B*a*b^2 - A*b^3)*d*e^3 + (3*B*a^2*b - 7*
A*a*b^2)*e^4)*x^3 + (4*B*b^3*d^3*e + (19*B*a*b^2 - 7*A*b^3)*d^2*e^2 + 4*(4*B*a^2*b - 7*A*a*b^2)*d*e^3 + (3*B*a
^3 - 7*A*a^2*b)*e^4)*x^2 + 2*(4*B*a*b^2*d^3*e + 7*(B*a^2*b - A*a*b^2)*d^2*e^2 + (3*B*a^3 - 7*A*a^2*b)*d*e^3)*x
)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (8*A*a^3*e^3 +
6*(B*a*b^2 + A*b^3)*d^3 + (83*B*a^2*b - 39*A*a*b^2)*d^2*e + 16*(B*a^3 - 5*A*a^2*b)*d*e^2 + 15*(4*B*b^3*d*e^2 +
(3*B*a*b^2 - 7*A*b^3)*e^3)*x^3 + 5*(16*B*b^3*d^2*e + 4*(8*B*a*b^2 - 7*A*b^3)*d*e^2 + 5*(3*B*a^2*b - 7*A*a*b^2
)*e^3)*x^2 + (12*B*b^3*d^3 + (145*B*a*b^2 - 21*A*b^3)*d^2*e + 2*(67*B*a^2*b - 119*A*a*b^2)*d*e^2 + 8*(3*B*a^3
- 7*A*a^2*b)*e^3)*x)*sqrt(e*x + d))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6
*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^
5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 -
9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2
*a^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*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)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.03677, size = 606, normalized size = 2.52 \begin{align*} -\frac{5 \,{\left (4 \, B b^{2} d e + 3 \, B a b e^{2} - 7 \, A b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (6 \,{\left (x e + d\right )} B b d e + B b d^{2} e + 3 \,{\left (x e + d\right )} B a e^{2} - 9 \,{\left (x e + d\right )} A b e^{2} - B a d e^{2} - A b d e^{2} + A a e^{3}\right )}}{3 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} d e - 4 \, \sqrt{x e + d} B b^{3} d^{2} e + 7 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{2} e^{2} - 11 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} e^{2} - 5 \, \sqrt{x e + d} B a b^{2} d e^{2} + 13 \, \sqrt{x e + d} A b^{3} d e^{2} + 9 \, \sqrt{x e + d} B a^{2} b e^{3} - 13 \, \sqrt{x e + d} A a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\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)^(5/2),x, algorithm="giac")

[Out]

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