3.2228 \(\int \frac{(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=257 \[ -\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.204377, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac{2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt{d+e x} (b d-a e)}+\frac{5 b (a+b x)^{3/2} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac{5 b \sqrt{a+b x} \sqrt{d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac{5 \sqrt{b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 e^{9/2}}-\frac{2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.148007, size = 113, normalized size = 0.44 \[ \frac{2 (a+b x)^{7/2} \left (-\frac{\left (\frac{b (d+e x)}{b d-a e}\right )^{3/2} (-3 a B e-4 A b e+7 b B d) \, _2F_1\left (\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{e (a+b x)}{a e-b d}\right )}{b}-7 A e+7 B d\right )}{21 e (d+e x)^{3/2} (a e-b d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a + b*x)^(7/2)*(7*B*d - 7*A*e - ((7*b*B*d - 4*A*b*e - 3*a*B*e)*((b*(d + e*x))/(b*d - a*e))^(3/2)*Hypergeom
etric2F1[3/2, 7/2, 9/2, (e*(a + b*x))/(-(b*d) + a*e)])/b))/(21*e*(-(b*d) + a*e)*(d + e*x)^(3/2))

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Maple [B]  time = 0.024, size = 1250, normalized size = 4.9 \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)^(5/2),x)

[Out]

1/24*(b*x+a)^(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^2*b^3*
d^2*e^2+24*A*x^2*b^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+210*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-150*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-48*B*x*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-32*B*a^2*d*e^2*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+54*B*x^2*a*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-42*B*x^2*b^2*d*e^2*((b*x+a)*(e*x+d))
^(1/2)*(b*e)^(1/2)-112*A*x*a*b*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-280*B*x*b^2*d^2*e*((b*x+a)*(e*x+d))^(1/
2)*(b*e)^(1/2)+12*B*x^3*b^2*e^3*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)-60*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-16*A*a^2*e^3*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+120*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+90*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+160*A*x*b^2*d*e^2*(b*e)^(1/2)*((b*x+a)*(e
*x+d))^(1/2)-80*A*a*b*d*e^2*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)+120*A*b^2*d^2*e*(b*e)^(1/2)*((b*x+a)*(e*x+d))^
(1/2)-120*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+230*B*a*
b*d^2*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+60*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^2*a*b^2*e^4-60*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^2*b^3*d*e^3+45*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^2*a^2*b*e^4+3
16*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+60*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)-150*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^2*a*b^2*d*e^3-300*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+45*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)^(3/2)/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 18.6963, size = 1877, normalized size = 7.3 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(15*(7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*a^2 + 4*A*a*b)*d^2*e^2 + (7*B*b^2*d^2*e^2 - 2*(5*B
*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x^2 + 2*(7*B*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B
*a^2 + 4*A*a*b)*d*e^3)*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^2*x + b*d*e +
a*e^2)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(b/e) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(6*B*b^2*e^3*x^3 - 105*B*b^2*d^3
- 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 3*(7*B*b^2*d*e^2 - (9*B*a*b + 4*
A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3)*x)*sqrt(b*x + a
)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^4), -1/24*(15*(7*B*b^2*d^4 - 2*(5*B*a*b + 2*A*b^2)*d^3*e + (3*B*
a^2 + 4*A*a*b)*d^2*e^2 + (7*B*b^2*d^2*e^2 - 2*(5*B*a*b + 2*A*b^2)*d*e^3 + (3*B*a^2 + 4*A*a*b)*e^4)*x^2 + 2*(7*
B*b^2*d^3*e - 2*(5*B*a*b + 2*A*b^2)*d^2*e^2 + (3*B*a^2 + 4*A*a*b)*d*e^3)*x)*sqrt(-b/e)*arctan(1/2*(2*b*e*x + b
*d + a*e)*sqrt(b*x + a)*sqrt(e*x + d)*sqrt(-b/e)/(b^2*e*x^2 + a*b*d + (b^2*d + a*b*e)*x)) - 2*(6*B*b^2*e^3*x^3
- 105*B*b^2*d^3 - 8*A*a^2*e^3 + 5*(23*B*a*b + 12*A*b^2)*d^2*e - 8*(2*B*a^2 + 5*A*a*b)*d*e^2 - 3*(7*B*b^2*d*e^
2 - (9*B*a*b + 4*A*b^2)*e^3)*x^2 - 2*(70*B*b^2*d^2*e - (79*B*a*b + 40*A*b^2)*d*e^2 + 4*(3*B*a^2 + 7*A*a*b)*e^3
)*x)*sqrt(b*x + a)*sqrt(e*x + d))/(e^6*x^2 + 2*d*e^5*x + d^2*e^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((b*x+a)**(5/2)*(B*x+A)/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.57477, size = 721, normalized size = 2.81 \begin{align*} -\frac{5 \,{\left (7 \, B b^{2} d^{2}{\left | b \right |} - 10 \, B a b d{\left | b \right |} e - 4 \, A b^{2} d{\left | b \right |} e + 3 \, B a^{2}{\left | b \right |} e^{2} + 4 \, A a b{\left | b \right |} e^{2}\right )} e^{\left (-\frac{9}{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 )}{4 \, \sqrt{b}} + \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (B b^{5} d{\left | b \right |} e^{6} - B a b^{4}{\left | b \right |} e^{7}\right )}{\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac{7 \, B b^{6} d^{2}{\left | b \right |} e^{5} - 10 \, B a b^{5} d{\left | b \right |} e^{6} - 4 \, A b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{2} b^{4}{\left | b \right |} e^{7} + 4 \, A a b^{5}{\left | b \right |} e^{7}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac{20 \,{\left (7 \, B b^{7} d^{3}{\left | b \right |} e^{4} - 17 \, B a b^{6} d^{2}{\left | b \right |} e^{5} - 4 \, A b^{7} d^{2}{\left | b \right |} e^{5} + 13 \, B a^{2} b^{5} d{\left | b \right |} e^{6} + 8 \, A a b^{6} d{\left | b \right |} e^{6} - 3 \, B a^{3} b^{4}{\left | b \right |} e^{7} - 4 \, A a^{2} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )}{\left (b x + a\right )} - \frac{15 \,{\left (7 \, B b^{8} d^{4}{\left | b \right |} e^{3} - 24 \, B a b^{7} d^{3}{\left | b \right |} e^{4} - 4 \, A b^{8} d^{3}{\left | b \right |} e^{4} + 30 \, B a^{2} b^{6} d^{2}{\left | b \right |} e^{5} + 12 \, A a b^{7} d^{2}{\left | b \right |} e^{5} - 16 \, B a^{3} b^{5} d{\left | b \right |} e^{6} - 12 \, A a^{2} b^{6} d{\left | b \right |} e^{6} + 3 \, B a^{4} b^{4}{\left | b \right |} e^{7} + 4 \, A a^{3} b^{5}{\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} d +{\left (b x + a\right )} b e - a b e\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

-5/4*(7*B*b^2*d^2*abs(b) - 10*B*a*b*d*abs(b)*e - 4*A*b^2*d*abs(b)*e + 3*B*a^2*abs(b)*e^2 + 4*A*a*b*abs(b)*e^2)
*e^(-9/2)*log(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) + 1/12*((3*(b
*x + a)*(2*(B*b^5*d*abs(b)*e^6 - B*a*b^4*abs(b)*e^7)*(b*x + a)/(b^4*d*e^7 - a*b^3*e^8) - (7*B*b^6*d^2*abs(b)*e
^5 - 10*B*a*b^5*d*abs(b)*e^6 - 4*A*b^6*d*abs(b)*e^6 + 3*B*a^2*b^4*abs(b)*e^7 + 4*A*a*b^5*abs(b)*e^7)/(b^4*d*e^
7 - a*b^3*e^8)) - 20*(7*B*b^7*d^3*abs(b)*e^4 - 17*B*a*b^6*d^2*abs(b)*e^5 - 4*A*b^7*d^2*abs(b)*e^5 + 13*B*a^2*b
^5*d*abs(b)*e^6 + 8*A*a*b^6*d*abs(b)*e^6 - 3*B*a^3*b^4*abs(b)*e^7 - 4*A*a^2*b^5*abs(b)*e^7)/(b^4*d*e^7 - a*b^3
*e^8))*(b*x + a) - 15*(7*B*b^8*d^4*abs(b)*e^3 - 24*B*a*b^7*d^3*abs(b)*e^4 - 4*A*b^8*d^3*abs(b)*e^4 + 30*B*a^2*
b^6*d^2*abs(b)*e^5 + 12*A*a*b^7*d^2*abs(b)*e^5 - 16*B*a^3*b^5*d*abs(b)*e^6 - 12*A*a^2*b^6*d*abs(b)*e^6 + 3*B*a
^4*b^4*abs(b)*e^7 + 4*A*a^3*b^5*abs(b)*e^7)/(b^4*d*e^7 - a*b^3*e^8))*sqrt(b*x + a)/(b^2*d + (b*x + a)*b*e - a*
b*e)^(3/2)