3.1758 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a+b x)^3} \, dx\)

Optimal. Leaf size=274 \[ -\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac{7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}+\frac{7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac{7 e \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}-\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d
+ e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d
+ 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d
- a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqr
t[b*d - a*e]])/(4*b^(11/2))

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Rubi [A]  time = 0.237184, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {78, 47, 50, 63, 208} \[ -\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 (a+b x) (b d-a e)}+\frac{7 e (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 (b d-a e)}+\frac{7 e (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4}+\frac{7 e \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5}-\frac{7 e (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}-\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x)^2 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Sqrt[d + e*x])/(4*b^5) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d
+ e*x)^(3/2))/(12*b^4) + (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)) - ((4*b*B*d
+ 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)) - ((A*b - a*B)*(d + e*x)^(9/2))/(2*b*(b*d
- a*e)*(a + b*x)^2) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqr
t[b*d - a*e]])/(4*b^(11/2))

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 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) (d+e x)^{7/2}}{(a+b x)^3} \, dx &=-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{(4 b B d+5 A b e-9 a B e) \int \frac{(d+e x)^{7/2}}{(a+b x)^2} \, dx}{4 b (b d-a e)}\\ &=-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{(7 e (4 b B d+5 A b e-9 a B e)) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{8 b^2 (b d-a e)}\\ &=\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{(7 e (4 b B d+5 A b e-9 a B e)) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{8 b^3}\\ &=\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{(7 e (b d-a e) (4 b B d+5 A b e-9 a B e)) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{8 b^4}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt{d+e x}}{4 b^5}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{\left (7 e (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 b^5}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt{d+e x}}{4 b^5}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}+\frac{\left (7 (b d-a e)^2 (4 b B d+5 A b e-9 a B e)\right ) \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^5}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) \sqrt{d+e x}}{4 b^5}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{3/2}}{12 b^4}+\frac{7 e (4 b B d+5 A b e-9 a B e) (d+e x)^{5/2}}{20 b^3 (b d-a e)}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x)^2}-\frac{7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0867159, size = 97, normalized size = 0.35 \[ \frac{(d+e x)^{9/2} \left (\frac{e (-9 a B e+5 A b e+4 b B d) \, _2F_1\left (2,\frac{9}{2};\frac{11}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+\frac{9 (a B-A b)}{(a+b x)^2}\right )}{18 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a + b*x)^3,x]

[Out]

((d + e*x)^(9/2)*((9*(-(A*b) + a*B))/(a + b*x)^2 + (e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*Hypergeometric2F1[2, 9/2,
11/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(18*b*(b*d - a*e))

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Maple [B]  time = 0.025, size = 940, normalized size = 3.4 \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)*(e*x+d)^(7/2)/(b*x+a)^3,x)

[Out]

-11/4/b^4/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^3*e^5+11/4/b/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*d^3*e^2+15/4/b^5/(b*e*x+a
*e)^2*(e*x+d)^(1/2)*B*a^4*e^5+35/4/b^4/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a^2*e
^4-18/b^4*B*a*d*e^2*(e*x+d)^(1/2)-e/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*d^3-2/b^4*B*(e*x+d)^(3/2)*a*e^2-6/b^4*A*a*
e^3*(e*x+d)^(1/2)+4/3*e/b^3*B*(e*x+d)^(3/2)*d+6*e/b^3*B*d^2*(e*x+d)^(1/2)+6/b^3*A*d*e^2*(e*x+d)^(1/2)-49/4/b^4
/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^3*d*e^4+35/4/b^2/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/
2))*A*d^2*e^2-63/4/b^5/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a^3*e^4+e/b/(b*e*x+a*
e)^2*(e*x+d)^(1/2)*B*d^4+7*e/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^3-13/4/b^
3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*a^2*A*e^4-13/4/b/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*d^2*e^2+17/4/b^4/(b*e*x+a*e)^2*(e
*x+d)^(3/2)*a^3*e^4*B-27/4/b^2/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a*d^3*e^2-35/2/b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(
e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*A*a*d*e^3+33/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*A*a^2*d*e^4-33/4/b^2/(b*e*x+a
*e)^2*(e*x+d)^(1/2)*A*a*d^2*e^3+13/2/b^2/(b*e*x+a*e)^2*(e*x+d)^(3/2)*A*a*d*e^3+12/b^5*a^2*e^3*B*(e*x+d)^(1/2)+
77/2/b^4/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a^2*d*e^3+25/4/b^2/(b*e*x+a*e)^2*(e
*x+d)^(3/2)*B*a*d^2*e^2-119/4/b^3/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*B*a*d^2*e^2-
19/2/b^3/(b*e*x+a*e)^2*(e*x+d)^(3/2)*B*a^2*d*e^3+57/4/b^3/(b*e*x+a*e)^2*(e*x+d)^(1/2)*B*a^2*d^2*e^3+2/5*e/b^3*
B*(e*x+d)^(5/2)+2/3/b^3*A*(e*x+d)^(3/2)*e^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((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.48919, size = 2276, normalized size = 8.31 \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)*(e*x+d)^(7/2)/(b*x+a)^3,x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^2*e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e
- (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*
A*a*b^3)*d*e^2 + (9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x +
d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(24*B*b^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15
*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (
9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b
^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 195*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*
(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(4*B*a^2*b^2*d^2*
e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e - (13*B*a*b^3 - 5*A*b^4)*d*e
^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*B*a^3*b -
5*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (24*B*b
^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*
d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e
- 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 19
5*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/
(b^7*x^2 + 2*a*b^6*x + a^2*b^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)*(e*x+d)**(7/2)/(b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 2.06435, size = 819, normalized size = 2.99 \begin{align*} \frac{7 \,{\left (4 \, B b^{3} d^{3} e - 17 \, B a b^{2} d^{2} e^{2} + 5 \, A b^{3} d^{2} e^{2} + 22 \, B a^{2} b d e^{3} - 10 \, A a b^{2} d e^{3} - 9 \, B a^{3} e^{4} + 5 \, A a^{2} b e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{5}} - \frac{4 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} d^{3} e - 4 \, \sqrt{x e + d} B b^{4} d^{4} e - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{3} d^{2} e^{2} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{4} d^{2} e^{2} + 27 \, \sqrt{x e + d} B a b^{3} d^{3} e^{2} - 11 \, \sqrt{x e + d} A b^{4} d^{3} e^{2} + 38 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{2} d e^{3} - 26 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{3} d e^{3} - 57 \, \sqrt{x e + d} B a^{2} b^{2} d^{2} e^{3} + 33 \, \sqrt{x e + d} A a b^{3} d^{2} e^{3} - 17 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{3} b e^{4} + 13 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2} b^{2} e^{4} + 49 \, \sqrt{x e + d} B a^{3} b d e^{4} - 33 \, \sqrt{x e + d} A a^{2} b^{2} d e^{4} - 15 \, \sqrt{x e + d} B a^{4} e^{5} + 11 \, \sqrt{x e + d} A a^{3} b e^{5}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{5}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B b^{12} e + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{12} d e + 45 \, \sqrt{x e + d} B b^{12} d^{2} e - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{11} e^{2} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{12} e^{2} - 135 \, \sqrt{x e + d} B a b^{11} d e^{2} + 45 \, \sqrt{x e + d} A b^{12} d e^{2} + 90 \, \sqrt{x e + d} B a^{2} b^{10} e^{3} - 45 \, \sqrt{x e + d} A a b^{11} e^{3}\right )}}{15 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b*x+a)^3,x, algorithm="giac")

[Out]

7/4*(4*B*b^3*d^3*e - 17*B*a*b^2*d^2*e^2 + 5*A*b^3*d^2*e^2 + 22*B*a^2*b*d*e^3 - 10*A*a*b^2*d*e^3 - 9*B*a^3*e^4
+ 5*A*a^2*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/4*(4*(x*e + d)^(3
/2)*B*b^4*d^3*e - 4*sqrt(x*e + d)*B*b^4*d^4*e - 25*(x*e + d)^(3/2)*B*a*b^3*d^2*e^2 + 13*(x*e + d)^(3/2)*A*b^4*
d^2*e^2 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^2 - 11*sqrt(x*e + d)*A*b^4*d^3*e^2 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e
^3 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^3 - 57*sqrt(x*e + d)*B*a^2*b^2*d^2*e^3 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^3
- 17*(x*e + d)^(3/2)*B*a^3*b*e^4 + 13*(x*e + d)^(3/2)*A*a^2*b^2*e^4 + 49*sqrt(x*e + d)*B*a^3*b*d*e^4 - 33*sqrt
(x*e + d)*A*a^2*b^2*d*e^4 - 15*sqrt(x*e + d)*B*a^4*e^5 + 11*sqrt(x*e + d)*A*a^3*b*e^5)/(((x*e + d)*b - b*d + a
*e)^2*b^5) + 2/15*(3*(x*e + d)^(5/2)*B*b^12*e + 10*(x*e + d)^(3/2)*B*b^12*d*e + 45*sqrt(x*e + d)*B*b^12*d^2*e
- 15*(x*e + d)^(3/2)*B*a*b^11*e^2 + 5*(x*e + d)^(3/2)*A*b^12*e^2 - 135*sqrt(x*e + d)*B*a*b^11*d*e^2 + 45*sqrt(
x*e + d)*A*b^12*d*e^2 + 90*sqrt(x*e + d)*B*a^2*b^10*e^3 - 45*sqrt(x*e + d)*A*a*b^11*e^3)/b^15