3.2227 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + (5*(b*d - a*e)*(7*b*B*d - 6*A*b*e - a*B*e)*Sq
rt[a + b*x]*Sqrt[d + e*x])/(8*e^4) - (5*(7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(12*e^3) +
((7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e^2*(b*d - a*e)) - (5*(b*d - a*e)^2*(7*b*B*d -
6*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*Sqrt[b]*e^(9/2))

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Rubi [A]  time = 0.210533, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {78, 50, 63, 217, 206} \[ \frac{(a+b x)^{5/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{3 e^2 (b d-a e)}-\frac{5 (a+b x)^{3/2} \sqrt{d+e x} (-a B e-6 A b e+7 b B d)}{12 e^3}+\frac{5 \sqrt{a+b x} \sqrt{d+e x} (b d-a e) (-a B e-6 A b e+7 b B d)}{8 e^4}-\frac{5 (b d-a e)^2 (-a B e-6 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}-\frac{2 (a+b x)^{7/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(B*d - A*e)*(a + b*x)^(7/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) + (5*(b*d - a*e)*(7*b*B*d - 6*A*b*e - a*B*e)*Sq
rt[a + b*x]*Sqrt[d + e*x])/(8*e^4) - (5*(7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(12*e^3) +
((7*b*B*d - 6*A*b*e - a*B*e)*(a + b*x)^(5/2)*Sqrt[d + e*x])/(3*e^2*(b*d - a*e)) - (5*(b*d - a*e)^2*(7*b*B*d -
6*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*Sqrt[b]*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 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{(a+b x)^{5/2} (A+B x)}{(d+e x)^{3/2}} \, dx &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{(7 b B d-6 A b e-a B e) \int \frac{(a+b x)^{5/2}}{\sqrt{d+e x}} \, dx}{e (b d-a e)}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}-\frac{(5 (7 b B d-6 A b e-a B e)) \int \frac{(a+b x)^{3/2}}{\sqrt{d+e x}} \, dx}{6 e^2}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}-\frac{5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{12 e^3}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}+\frac{(5 (b d-a e) (7 b B d-6 A b e-a B e)) \int \frac{\sqrt{a+b x}}{\sqrt{d+e x}} \, dx}{8 e^3}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 e^4}-\frac{5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{12 e^3}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}-\frac{\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{d+e x}} \, dx}{16 e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 e^4}-\frac{5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{12 e^3}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}-\frac{\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{a e}{b}+\frac{e x^2}{b}}} \, dx,x,\sqrt{a+b x}\right )}{8 b e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 e^4}-\frac{5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{12 e^3}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}-\frac{\left (5 (b d-a e)^2 (7 b B d-6 A b e-a B e)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{b}} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{d+e x}}\right )}{8 b e^4}\\ &=-\frac{2 (B d-A e) (a+b x)^{7/2}}{e (b d-a e) \sqrt{d+e x}}+\frac{5 (b d-a e) (7 b B d-6 A b e-a B e) \sqrt{a+b x} \sqrt{d+e x}}{8 e^4}-\frac{5 (7 b B d-6 A b e-a B e) (a+b x)^{3/2} \sqrt{d+e x}}{12 e^3}+\frac{(7 b B d-6 A b e-a B e) (a+b x)^{5/2} \sqrt{d+e x}}{3 e^2 (b d-a e)}-\frac{5 (b d-a e)^2 (7 b B d-6 A b e-a B e) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 \sqrt{b} e^{9/2}}\\ \end{align*}

Mathematica [A]  time = 1.28199, size = 309, normalized size = 1.22 \[ \frac{\frac{\sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}} (a B e+6 A b e-7 b B d) \left (8 b^3 e^3 (a+b x)^3 \sqrt{b d-a e} \sqrt{\frac{b (d+e x)}{b d-a e}}-10 b^3 e^2 (a+b x)^2 (b d-a e)^{3/2} \sqrt{\frac{b (d+e x)}{b d-a e}}+15 b^3 e (a+b x) (b d-a e)^{5/2} \sqrt{\frac{b (d+e x)}{b d-a e}}-15 b^3 \sqrt{e} \sqrt{a+b x} (b d-a e)^3 \sinh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b d-a e}}\right )\right )}{3 b^4}+16 e^4 (a+b x)^4 (B d-A e)}{8 e^5 \sqrt{a+b x} \sqrt{d+e x} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*e^4*(B*d - A*e)*(a + b*x)^4 + (Sqrt[b*d - a*e]*(-7*b*B*d + 6*A*b*e + a*B*e)*Sqrt[(b*(d + e*x))/(b*d - a*e)
]*(15*b^3*e*(b*d - a*e)^(5/2)*(a + b*x)*Sqrt[(b*(d + e*x))/(b*d - a*e)] - 10*b^3*e^2*(b*d - a*e)^(3/2)*(a + b*
x)^2*Sqrt[(b*(d + e*x))/(b*d - a*e)] + 8*b^3*e^3*Sqrt[b*d - a*e]*(a + b*x)^3*Sqrt[(b*(d + e*x))/(b*d - a*e)] -
15*b^3*Sqrt[e]*(b*d - a*e)^3*Sqrt[a + b*x]*ArcSinh[(Sqrt[e]*Sqrt[a + b*x])/Sqrt[b*d - a*e]]))/(3*b^4))/(8*e^5
*(-(b*d) + a*e)*Sqrt[a + b*x]*Sqrt[d + e*x])

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Maple [B]  time = 0.029, size = 1184, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(b*x+a)^(1/2)*(24*A*x^2*b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^3*d^3*e+225*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d^3*e+66*B*x*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+162*B*a^2*d*e^2*((
b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+52*B*x^2*a*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-28*B*x^2*b^2*d*e^2*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+108*A*x*a*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+70*B*x*b^2*d^2*e*((b*x+a)*
(e*x+d))^(1/2)*(b*e)^(1/2)+16*B*x^3*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+90*A*ln(1/2*(2*b*x*e+2*((b*x+a
)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^3*d^3*e+15*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b
*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3*d*e^3-96*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+15*B*ln(1/2*(2*b*x*
e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^3*e^4+90*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b*d*e^3-180*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(
1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^2*d*e^3-135*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/
(b*e)^(1/2))*x*a^2*b*d*e^3-60*A*x*b^2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+300*A*a*b*d*e^2*(b*e)^(1/2)*((
b*x+a)*(e*x+d))^(1/2)-180*A*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+90*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x
+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*b*e^4+90*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^
(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^3*d^2*e^2-380*B*a*b*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-136*B*x*a*b*d*e^
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-105*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*
e)^(1/2))*b^3*d^4-180*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b^2*d^2*
e^2+210*B*b^2*d^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+225*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1
/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b^2*d^2*e^2-135*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)
/(b*e)^(1/2))*a^2*b*d^2*e^2)/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+d)^(1/2)/e^4

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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)^(5/2)*(B*x+A)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.15423, size = 1890, normalized size = 7.47 \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)^(5/2)*(B*x+A)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

[-1/96*(15*(7*B*b^3*d^4 - 3*(5*B*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2
*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 + 3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*
a^2*b)*e^4)*x)*sqrt(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 + 4*(2*b*e*x + b*d + a*e)*sqrt(b*e)
*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) - 4*(8*B*b^3*e^4*x^3 + 105*B*b^3*d^3*e - 48*A*a^2*b*e^
4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2*(7*B*b^3*d*e^3 - (13*B*a*b^2 + 6
*A*b^3)*e^4)*x^2 + (35*B*b^3*d^2*e^2 - 2*(34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a^2*b + 18*A*a*b^2)*e^4)*x)*s
qrt(b*x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5), 1/48*(15*(7*B*b^3*d^4 - 3*(5*B*a*b^2 + 2*A*b^3)*d^3*e + 3*(3*
B*a^2*b + 4*A*a*b^2)*d^2*e^2 - (B*a^3 + 6*A*a^2*b)*d*e^3 + (7*B*b^3*d^3*e - 3*(5*B*a*b^2 + 2*A*b^3)*d^2*e^2 +
3*(3*B*a^2*b + 4*A*a*b^2)*d*e^3 - (B*a^3 + 6*A*a^2*b)*e^4)*x)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt
(-b*e)*sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x)) + 2*(8*B*b^3*e^4*x^3 + 105
*B*b^3*d^3*e - 48*A*a^2*b*e^4 - 10*(19*B*a*b^2 + 9*A*b^3)*d^2*e^2 + 3*(27*B*a^2*b + 50*A*a*b^2)*d*e^3 - 2*(7*B
*b^3*d*e^3 - (13*B*a*b^2 + 6*A*b^3)*e^4)*x^2 + (35*B*b^3*d^2*e^2 - 2*(34*B*a*b^2 + 15*A*b^3)*d*e^3 + 3*(11*B*a
^2*b + 18*A*a*b^2)*e^4)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(b*e^6*x + b*d*e^5)]

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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)**(5/2)*(B*x+A)/(e*x+d)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 2.01693, size = 564, normalized size = 2.23 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \,{\left (b x + a\right )} B b{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}} - \frac{7 \, B b^{2} d{\left | b \right |} e^{5} - B a b{\left | b \right |} e^{6} - 6 \, A b^{2}{\left | b \right |} e^{6}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{5 \,{\left (7 \, B b^{3} d^{2}{\left | b \right |} e^{4} - 8 \, B a b^{2} d{\left | b \right |} e^{5} - 6 \, A b^{3} d{\left | b \right |} e^{5} + B a^{2} b{\left | b \right |} e^{6} + 6 \, A a b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )}{\left (b x + a\right )} + \frac{15 \,{\left (7 \, B b^{4} d^{3}{\left | b \right |} e^{3} - 15 \, B a b^{3} d^{2}{\left | b \right |} e^{4} - 6 \, A b^{4} d^{2}{\left | b \right |} e^{4} + 9 \, B a^{2} b^{2} d{\left | b \right |} e^{5} + 12 \, A a b^{3} d{\left | b \right |} e^{5} - B a^{3} b{\left | b \right |} e^{6} - 6 \, A a^{2} b^{2}{\left | b \right |} e^{6}\right )}}{b^{10} d e^{8} - a b^{9} e^{9}}\right )} \sqrt{b x + a}}{184320 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} + \frac{{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 8 \, B a b d{\left | b \right |} e - 6 \, A b^{2} d{\left | b \right |} e + B a^{2}{\left | b \right |} e^{2} + 6 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{11}{2}\right )} \log \left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{12288 \, b^{\frac{17}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/184320*((2*(4*(b*x + a)*B*b*abs(b)*e^6/(b^10*d*e^8 - a*b^9*e^9) - (7*B*b^2*d*abs(b)*e^5 - B*a*b*abs(b)*e^6 -
6*A*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*(b*x + a) + 5*(7*B*b^3*d^2*abs(b)*e^4 - 8*B*a*b^2*d*abs(b)*e^5
- 6*A*b^3*d*abs(b)*e^5 + B*a^2*b*abs(b)*e^6 + 6*A*a*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*(b*x + a) + 15*(
7*B*b^4*d^3*abs(b)*e^3 - 15*B*a*b^3*d^2*abs(b)*e^4 - 6*A*b^4*d^2*abs(b)*e^4 + 9*B*a^2*b^2*d*abs(b)*e^5 + 12*A*
a*b^3*d*abs(b)*e^5 - B*a^3*b*abs(b)*e^6 - 6*A*a^2*b^2*abs(b)*e^6)/(b^10*d*e^8 - a*b^9*e^9))*sqrt(b*x + a)/sqrt
(b^2*d + (b*x + a)*b*e - a*b*e) + 1/12288*(7*B*b^2*d^2*abs(b) - 8*B*a*b*d*abs(b)*e - 6*A*b^2*d*abs(b)*e + B*a^
2*abs(b)*e^2 + 6*A*a*b*abs(b)*e^2)*e^(-11/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b
*e - a*b*e)))/b^(17/2)