3.18.91 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^{7/2}} \, dx\) [1791]

Optimal. Leaf size=124 \[ \frac {2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}-\frac {2 (b d-a e) (3 b B d-2 A b e-a B e)}{3 e^4 (d+e x)^{3/2}}+\frac {2 b (3 b B d-A b e-2 a B e)}{e^4 \sqrt {d+e x}}+\frac {2 b^2 B \sqrt {d+e x}}{e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 78} \begin {gather*} \frac {2 b (-2 a B e-A b e+3 b B d)}{e^4 \sqrt {d+e x}}-\frac {2 (b d-a e) (-a B e-2 A b e+3 b B d)}{3 e^4 (d+e x)^{3/2}}+\frac {2 (b d-a e)^2 (B d-A e)}{5 e^4 (d+e x)^{5/2}}+\frac {2 b^2 B \sqrt {d+e x}}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 136, normalized size = 1.10 \begin {gather*} -\frac {2 \left (a^2 e^2 (2 B d+3 A e+5 B e x)+2 a b e \left (A e (2 d+5 e x)+B \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )+b^2 \left (A e \left (8 d^2+20 d e x+15 e^2 x^2\right )-3 B \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{15 e^4 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a^2*e^2*(2*B*d + 3*A*e + 5*B*e*x) + 2*a*b*e*(A*e*(2*d + 5*e*x) + B*(8*d^2 + 20*d*e*x + 15*e^2*x^2)) + b^2
*(A*e*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 3*B*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3))))/(15*e^4*(d + e
*x)^(5/2))

________________________________________________________________________________________

Maple [A]
time = 0.70, size = 158, normalized size = 1.27

method	result	size
derivativedivides	\(\frac {2 b^{2} B \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +a^{2} B \,e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b \left (A b e +2 B a e -3 B b d \right )}{\sqrt {e x +d}}}{e^{4}}\)	\(158\)
default	\(\frac {2 b^{2} B \sqrt {e x +d}-\frac {2 \left (2 A a b \,e^{2}-2 A \,b^{2} d e +a^{2} B \,e^{2}-4 B a b d e +3 b^{2} B \,d^{2}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (a^{2} A \,e^{3}-2 A a b d \,e^{2}+A \,b^{2} d^{2} e -B \,a^{2} d \,e^{2}+2 B a b \,d^{2} e -b^{2} B \,d^{3}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {2 b \left (A b e +2 B a e -3 B b d \right )}{\sqrt {e x +d}}}{e^{4}}\)	\(158\)
gosper	\(-\frac {2 \left (-15 b^{2} B \,x^{3} e^{3}+15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-90 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -120 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\)	\(169\)
trager	\(-\frac {2 \left (-15 b^{2} B \,x^{3} e^{3}+15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-90 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -120 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -48 b^{2} B \,d^{3}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\)	\(169\)
risch	\(\frac {2 b^{2} B \sqrt {e x +d}}{e^{4}}-\frac {2 \left (15 A \,b^{2} e^{3} x^{2}+30 B a b \,e^{3} x^{2}-45 B \,b^{2} d \,e^{2} x^{2}+10 A a b \,e^{3} x +20 A \,b^{2} d \,e^{2} x +5 B \,a^{2} e^{3} x +40 B a b d \,e^{2} x -75 B \,b^{2} d^{2} e x +3 a^{2} A \,e^{3}+4 A a b d \,e^{2}+8 A \,b^{2} d^{2} e +2 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -33 b^{2} B \,d^{3}\right )}{15 e^{4} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d x e +d^{2}\right )}\)	\(192\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 171, normalized size = 1.38 \begin {gather*} \frac {2}{15} \, {\left (15 \, \sqrt {x e + d} B b^{2} e^{\left (-3\right )} + \frac {{\left (3 \, B b^{2} d^{3} - 3 \, A a^{2} e^{3} + 15 \, {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} {\left (x e + d\right )}^{2} - 3 \, {\left (2 \, B a b e + A b^{2} e\right )} d^{2} - 5 \, {\left (3 \, B b^{2} d^{2} + B a^{2} e^{2} + 2 \, A a b e^{2} - 2 \, {\left (2 \, B a b e + A b^{2} e\right )} d\right )} {\left (x e + d\right )} + 3 \, {\left (B a^{2} e^{2} + 2 \, A a b e^{2}\right )} d\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{\frac {5}{2}}}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(15*sqrt(x*e + d)*B*b^2*e^(-3) + (3*B*b^2*d^3 - 3*A*a^2*e^3 + 15*(3*B*b^2*d - 2*B*a*b*e - A*b^2*e)*(x*e +
d)^2 - 3*(2*B*a*b*e + A*b^2*e)*d^2 - 5*(3*B*b^2*d^2 + B*a^2*e^2 + 2*A*a*b*e^2 - 2*(2*B*a*b*e + A*b^2*e)*d)*(x
*e + d) + 3*(B*a^2*e^2 + 2*A*a*b*e^2)*d)*e^(-3)/(x*e + d)^(5/2))*e^(-1)

________________________________________________________________________________________

Fricas [A]
time = 1.08, size = 175, normalized size = 1.41 \begin {gather*} \frac {2 \, {\left (48 \, B b^{2} d^{3} + {\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )} e^{3} + 2 \, {\left (45 \, B b^{2} d x^{2} - 10 \, {\left (2 \, B a b + A b^{2}\right )} d x - {\left (B a^{2} + 2 \, A a b\right )} d\right )} e^{2} + 8 \, {\left (15 \, B b^{2} d^{2} x - {\left (2 \, B a b + A b^{2}\right )} d^{2}\right )} e\right )} \sqrt {x e + d}}{15 \, {\left (x^{3} e^{7} + 3 \, d x^{2} e^{6} + 3 \, d^{2} x e^{5} + d^{3} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(48*B*b^2*d^3 + (15*B*b^2*x^3 - 3*A*a^2 - 15*(2*B*a*b + A*b^2)*x^2 - 5*(B*a^2 + 2*A*a*b)*x)*e^3 + 2*(45*B
*b^2*d*x^2 - 10*(2*B*a*b + A*b^2)*d*x - (B*a^2 + 2*A*a*b)*d)*e^2 + 8*(15*B*b^2*d^2*x - (2*B*a*b + A*b^2)*d^2)*
e)*sqrt(x*e + d)/(x^3*e^7 + 3*d*x^2*e^6 + 3*d^2*x*e^5 + d^3*e^4)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (126) = 252\).
time = 0.68, size = 1015, normalized size = 8.19 \begin {gather*} \begin {cases} - \frac {6 A a^{2} e^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {8 A a b d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {20 A a b e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 A b^{2} d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 A b^{2} d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 A b^{2} e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 B a^{2} d e^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 B a^{2} e^{3} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {32 B a b d^{2} e}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {80 B a b d e^{2} x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} - \frac {60 B a b e^{3} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {96 B b^{2} d^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {240 B b^{2} d^{2} e x}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {180 B b^{2} d e^{2} x^{2}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} + \frac {30 B b^{2} e^{3} x^{3}}{15 d^{2} e^{4} \sqrt {d + e x} + 30 d e^{5} x \sqrt {d + e x} + 15 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**2*e**3/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)
) - 8*A*a*b*d*e**2/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 20*
A*a*b*e**3*x/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 16*A*b**2
*d**2*e/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 40*A*b**2*d*e*
*2*x/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 30*A*b**2*e**3*x*
*2/(15*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 4*B*a**2*d*e**2/(15
*d**2*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 10*B*a**2*e**3*x/(15*d**2
*e**4*sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 32*B*a*b*d**2*e/(15*d**2*e**4*
sqrt(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 80*B*a*b*d*e**2*x/(15*d**2*e**4*sqrt
(d + e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) - 60*B*a*b*e**3*x**2/(15*d**2*e**4*sqrt(d
+ e*x) + 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 96*B*b**2*d**3/(15*d**2*e**4*sqrt(d + e*x)
+ 30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 240*B*b**2*d**2*e*x/(15*d**2*e**4*sqrt(d + e*x) +
30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 180*B*b**2*d*e**2*x**2/(15*d**2*e**4*sqrt(d + e*x) +
30*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)) + 30*B*b**2*e**3*x**3/(15*d**2*e**4*sqrt(d + e*x) + 3
0*d*e**5*x*sqrt(d + e*x) + 15*e**6*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B
*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4)/d**(7/2), True))

________________________________________________________________________________________

Giac [A]
time = 1.07, size = 202, normalized size = 1.63 \begin {gather*} 2 \, \sqrt {x e + d} B b^{2} e^{\left (-4\right )} + \frac {2 \, {\left (45 \, {\left (x e + d\right )}^{2} B b^{2} d - 15 \, {\left (x e + d\right )} B b^{2} d^{2} + 3 \, B b^{2} d^{3} - 30 \, {\left (x e + d\right )}^{2} B a b e - 15 \, {\left (x e + d\right )}^{2} A b^{2} e + 20 \, {\left (x e + d\right )} B a b d e + 10 \, {\left (x e + d\right )} A b^{2} d e - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e - 5 \, {\left (x e + d\right )} B a^{2} e^{2} - 10 \, {\left (x e + d\right )} A a b e^{2} + 3 \, B a^{2} d e^{2} + 6 \, A a b d e^{2} - 3 \, A a^{2} e^{3}\right )} e^{\left (-4\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)*B*b^2*e^(-4) + 2/15*(45*(x*e + d)^2*B*b^2*d - 15*(x*e + d)*B*b^2*d^2 + 3*B*b^2*d^3 - 30*(x*e +
d)^2*B*a*b*e - 15*(x*e + d)^2*A*b^2*e + 20*(x*e + d)*B*a*b*d*e + 10*(x*e + d)*A*b^2*d*e - 6*B*a*b*d^2*e - 3*A
*b^2*d^2*e - 5*(x*e + d)*B*a^2*e^2 - 10*(x*e + d)*A*a*b*e^2 + 3*B*a^2*d*e^2 + 6*A*a*b*d*e^2 - 3*A*a^2*e^3)*e^(
-4)/(x*e + d)^(5/2)

________________________________________________________________________________________

Mupad [B]
time = 0.10, size = 168, normalized size = 1.35 \begin {gather*} -\frac {2\,\left (2\,B\,a^2\,d\,e^2+5\,B\,a^2\,e^3\,x+3\,A\,a^2\,e^3+16\,B\,a\,b\,d^2\,e+40\,B\,a\,b\,d\,e^2\,x+4\,A\,a\,b\,d\,e^2+30\,B\,a\,b\,e^3\,x^2+10\,A\,a\,b\,e^3\,x-48\,B\,b^2\,d^3-120\,B\,b^2\,d^2\,e\,x+8\,A\,b^2\,d^2\,e-90\,B\,b^2\,d\,e^2\,x^2+20\,A\,b^2\,d\,e^2\,x-15\,B\,b^2\,e^3\,x^3+15\,A\,b^2\,e^3\,x^2\right )}{15\,e^4\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x)^(7/2),x)

[Out]

-(2*(3*A*a^2*e^3 - 48*B*b^2*d^3 + 8*A*b^2*d^2*e + 2*B*a^2*d*e^2 + 5*B*a^2*e^3*x + 15*A*b^2*e^3*x^2 - 15*B*b^2*
e^3*x^3 + 30*B*a*b*e^3*x^2 + 20*A*b^2*d*e^2*x - 120*B*b^2*d^2*e*x - 90*B*b^2*d*e^2*x^2 + 4*A*a*b*d*e^2 + 16*B*
a*b*d^2*e + 10*A*a*b*e^3*x + 40*B*a*b*d*e^2*x))/(15*e^4*(d + e*x)^(5/2))

________________________________________________________________________________________