3.1799 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=214 \[ -\frac {2 b^3 (d+e x)^{3/2} (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac {4 b^2 \sqrt {d+e x} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac {4 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 \sqrt {d+e x}}-\frac {2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^{3/2}}+\frac {2 (b d-a e)^4 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac {2 b^4 B (d+e x)^{5/2}}{5 e^6} \]

[Out]

2/5*(-a*e+b*d)^4*(-A*e+B*d)/e^6/(e*x+d)^(5/2)-2/3*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)/e^6/(e*x+d)^(3/2)-2/3*
b^3*(-A*b*e-4*B*a*e+5*B*b*d)*(e*x+d)^(3/2)/e^6+2/5*b^4*B*(e*x+d)^(5/2)/e^6+4*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+
5*B*b*d)/e^6/(e*x+d)^(1/2)+4*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*x+d)^(1/2)/e^6

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Rubi [A]  time = 0.09, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 77} \[ -\frac {2 b^3 (d+e x)^{3/2} (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac {4 b^2 \sqrt {d+e x} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac {4 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 \sqrt {d+e x}}-\frac {2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6 (d+e x)^{3/2}}+\frac {2 (b d-a e)^4 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac {2 b^4 B (d+e x)^{5/2}}{5 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 77

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 )^2}{(d+e x)^{7/2}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e)}{e^5 (d+e x)^{7/2}}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e)}{e^5 (d+e x)^{5/2}}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e)}{e^5 (d+e x)^{3/2}}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e)}{e^5 \sqrt {d+e x}}+\frac {b^3 (-5 b B d+A b e+4 a B e) \sqrt {d+e x}}{e^5}+\frac {b^4 B (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac {2 (b d-a e)^4 (B d-A e)}{5 e^6 (d+e x)^{5/2}}-\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e)}{3 e^6 (d+e x)^{3/2}}+\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{e^6 \sqrt {d+e x}}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) \sqrt {d+e x}}{e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{3/2}}{3 e^6}+\frac {2 b^4 B (d+e x)^{5/2}}{5 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 183, normalized size = 0.86 \[ \frac {2 \left (-5 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+30 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)+30 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)-5 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)+3 (b d-a e)^4 (B d-A e)+3 b^4 B (d+e x)^5\right )}{15 e^6 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.93, size = 441, normalized size = 2.06 \[ \frac {2 \, {\left (3 \, B b^{4} e^{5} x^{5} + 256 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} - 128 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 96 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 16 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 2 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \, {\left (2 \, B b^{4} d e^{4} - {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 10 \, {\left (8 \, B b^{4} d^{2} e^{3} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 30 \, {\left (16 \, B b^{4} d^{3} e^{2} - 8 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \, {\left (128 \, B b^{4} d^{4} e - 64 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 48 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 8 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

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)^2/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*B*b^4*e^5*x^5 + 256*B*b^4*d^5 - 3*A*a^4*e^5 - 128*(4*B*a*b^3 + A*b^4)*d^4*e + 96*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^3*e^2 - 16*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - 2*(B*a^4 + 4*A*a^3*b)*d*e^4 - 5*(2*B*b^4*d*e^4 - (4*B*a*
b^3 + A*b^4)*e^5)*x^4 + 10*(8*B*b^4*d^2*e^3 - 4*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x
^3 + 30*(16*B*b^4*d^3*e^2 - 8*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - (2*B*a^3*b + 3
*A*a^2*b^2)*e^5)*x^2 + 5*(128*B*b^4*d^4*e - 64*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 48*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*
e^3 - 8*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A*a^3*b)*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3
*d^2*e^7*x + d^3*e^6)

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giac [B]  time = 0.25, size = 567, normalized size = 2.65 \[ \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} e^{24} - 25 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d e^{24} + 150 \, \sqrt {x e + d} B b^{4} d^{2} e^{24} + 20 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} e^{25} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} e^{25} - 240 \, \sqrt {x e + d} B a b^{3} d e^{25} - 60 \, \sqrt {x e + d} A b^{4} d e^{25} + 90 \, \sqrt {x e + d} B a^{2} b^{2} e^{26} + 60 \, \sqrt {x e + d} A a b^{3} e^{26}\right )} e^{\left (-30\right )} + \frac {2 \, {\left (150 \, {\left (x e + d\right )}^{2} B b^{4} d^{3} - 25 \, {\left (x e + d\right )} B b^{4} d^{4} + 3 \, B b^{4} d^{5} - 360 \, {\left (x e + d\right )}^{2} B a b^{3} d^{2} e - 90 \, {\left (x e + d\right )}^{2} A b^{4} d^{2} e + 80 \, {\left (x e + d\right )} B a b^{3} d^{3} e + 20 \, {\left (x e + d\right )} A b^{4} d^{3} e - 12 \, B a b^{3} d^{4} e - 3 \, A b^{4} d^{4} e + 270 \, {\left (x e + d\right )}^{2} B a^{2} b^{2} d e^{2} + 180 \, {\left (x e + d\right )}^{2} A a b^{3} d e^{2} - 90 \, {\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{2} - 60 \, {\left (x e + d\right )} A a b^{3} d^{2} e^{2} + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} - 60 \, {\left (x e + d\right )}^{2} B a^{3} b e^{3} - 90 \, {\left (x e + d\right )}^{2} A a^{2} b^{2} e^{3} + 40 \, {\left (x e + d\right )} B a^{3} b d e^{3} + 60 \, {\left (x e + d\right )} A a^{2} b^{2} d e^{3} - 12 \, B a^{3} b d^{2} e^{3} - 18 \, A a^{2} b^{2} d^{2} e^{3} - 5 \, {\left (x e + d\right )} B a^{4} e^{4} - 20 \, {\left (x e + d\right )} A a^{3} b e^{4} + 3 \, B a^{4} d e^{4} + 12 \, A a^{3} b d e^{4} - 3 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \]

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)^2/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*b^4*e^24 - 25*(x*e + d)^(3/2)*B*b^4*d*e^24 + 150*sqrt(x*e + d)*B*b^4*d^2*e^24 + 20*(
x*e + d)^(3/2)*B*a*b^3*e^25 + 5*(x*e + d)^(3/2)*A*b^4*e^25 - 240*sqrt(x*e + d)*B*a*b^3*d*e^25 - 60*sqrt(x*e +
d)*A*b^4*d*e^25 + 90*sqrt(x*e + d)*B*a^2*b^2*e^26 + 60*sqrt(x*e + d)*A*a*b^3*e^26)*e^(-30) + 2/15*(150*(x*e +
d)^2*B*b^4*d^3 - 25*(x*e + d)*B*b^4*d^4 + 3*B*b^4*d^5 - 360*(x*e + d)^2*B*a*b^3*d^2*e - 90*(x*e + d)^2*A*b^4*d
^2*e + 80*(x*e + d)*B*a*b^3*d^3*e + 20*(x*e + d)*A*b^4*d^3*e - 12*B*a*b^3*d^4*e - 3*A*b^4*d^4*e + 270*(x*e + d
)^2*B*a^2*b^2*d*e^2 + 180*(x*e + d)^2*A*a*b^3*d*e^2 - 90*(x*e + d)*B*a^2*b^2*d^2*e^2 - 60*(x*e + d)*A*a*b^3*d^
2*e^2 + 18*B*a^2*b^2*d^3*e^2 + 12*A*a*b^3*d^3*e^2 - 60*(x*e + d)^2*B*a^3*b*e^3 - 90*(x*e + d)^2*A*a^2*b^2*e^3
+ 40*(x*e + d)*B*a^3*b*d*e^3 + 60*(x*e + d)*A*a^2*b^2*d*e^3 - 12*B*a^3*b*d^2*e^3 - 18*A*a^2*b^2*d^2*e^3 - 5*(x
*e + d)*B*a^4*e^4 - 20*(x*e + d)*A*a^3*b*e^4 + 3*B*a^4*d*e^4 + 12*A*a^3*b*d*e^4 - 3*A*a^4*e^5)*e^(-6)/(x*e + d
)^(5/2)

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maple [B]  time = 0.05, size = 469, normalized size = 2.19 \[ -\frac {2 \left (-3 b^{4} B \,x^{5} e^{5}-5 A \,b^{4} e^{5} x^{4}-20 B a \,b^{3} e^{5} x^{4}+10 B \,b^{4} d \,e^{4} x^{4}-60 A a \,b^{3} e^{5} x^{3}+40 A \,b^{4} d \,e^{4} x^{3}-90 B \,a^{2} b^{2} e^{5} x^{3}+160 B a \,b^{3} d \,e^{4} x^{3}-80 B \,b^{4} d^{2} e^{3} x^{3}+90 A \,a^{2} b^{2} e^{5} x^{2}-360 A a \,b^{3} d \,e^{4} x^{2}+240 A \,b^{4} d^{2} e^{3} x^{2}+60 B \,a^{3} b \,e^{5} x^{2}-540 B \,a^{2} b^{2} d \,e^{4} x^{2}+960 B a \,b^{3} d^{2} e^{3} x^{2}-480 B \,b^{4} d^{3} e^{2} x^{2}+20 A \,a^{3} b \,e^{5} x +120 A \,a^{2} b^{2} d \,e^{4} x -480 A a \,b^{3} d^{2} e^{3} x +320 A \,b^{4} d^{3} e^{2} x +5 B \,a^{4} e^{5} x +80 B \,a^{3} b d \,e^{4} x -720 B \,a^{2} b^{2} d^{2} e^{3} x +1280 B a \,b^{3} d^{3} e^{2} x -640 B \,b^{4} d^{4} e x +3 A \,a^{4} e^{5}+8 A \,a^{3} b d \,e^{4}+48 A \,a^{2} b^{2} d^{2} e^{3}-192 A a \,b^{3} d^{3} e^{2}+128 A \,b^{4} d^{4} e +2 B \,a^{4} d \,e^{4}+32 B \,d^{2} a^{3} b \,e^{3}-288 B \,d^{3} a^{2} b^{2} e^{2}+512 B a \,b^{3} d^{4} e -256 B \,b^{4} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}} \]

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)^2/(e*x+d)^(7/2),x)

[Out]

-2/15*(-3*B*b^4*e^5*x^5-5*A*b^4*e^5*x^4-20*B*a*b^3*e^5*x^4+10*B*b^4*d*e^4*x^4-60*A*a*b^3*e^5*x^3+40*A*b^4*d*e^
4*x^3-90*B*a^2*b^2*e^5*x^3+160*B*a*b^3*d*e^4*x^3-80*B*b^4*d^2*e^3*x^3+90*A*a^2*b^2*e^5*x^2-360*A*a*b^3*d*e^4*x
^2+240*A*b^4*d^2*e^3*x^2+60*B*a^3*b*e^5*x^2-540*B*a^2*b^2*d*e^4*x^2+960*B*a*b^3*d^2*e^3*x^2-480*B*b^4*d^3*e^2*
x^2+20*A*a^3*b*e^5*x+120*A*a^2*b^2*d*e^4*x-480*A*a*b^3*d^2*e^3*x+320*A*b^4*d^3*e^2*x+5*B*a^4*e^5*x+80*B*a^3*b*
d*e^4*x-720*B*a^2*b^2*d^2*e^3*x+1280*B*a*b^3*d^3*e^2*x-640*B*b^4*d^4*e*x+3*A*a^4*e^5+8*A*a^3*b*d*e^4+48*A*a^2*
b^2*d^2*e^3-192*A*a*b^3*d^3*e^2+128*A*b^4*d^4*e+2*B*a^4*d*e^4+32*B*a^3*b*d^2*e^3-288*B*a^2*b^2*d^3*e^2+512*B*a
*b^3*d^4*e-256*B*b^4*d^5)/(e*x+d)^(5/2)/e^6

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maxima [B]  time = 0.65, size = 416, normalized size = 1.94 \[ \frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{4} - 5 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 30 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 6 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 30 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \]

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)^2/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.00, size = 413, normalized size = 1.93 \[ \frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{3\,e^6}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^4\,e^4}{3}-\frac {16\,B\,a^3\,b\,d\,e^3}{3}+\frac {8\,A\,a^3\,b\,e^4}{3}+12\,B\,a^2\,b^2\,d^2\,e^2-8\,A\,a^2\,b^2\,d\,e^3-\frac {32\,B\,a\,b^3\,d^3\,e}{3}+8\,A\,a\,b^3\,d^2\,e^2+\frac {10\,B\,b^4\,d^4}{3}-\frac {8\,A\,b^4\,d^3\,e}{3}\right )+{\left (d+e\,x\right )}^2\,\left (8\,B\,a^3\,b\,e^3-36\,B\,a^2\,b^2\,d\,e^2+12\,A\,a^2\,b^2\,e^3+48\,B\,a\,b^3\,d^2\,e-24\,A\,a\,b^3\,d\,e^2-20\,B\,b^4\,d^3+12\,A\,b^4\,d^2\,e\right )+\frac {2\,A\,a^4\,e^5}{5}-\frac {2\,B\,b^4\,d^5}{5}+\frac {2\,A\,b^4\,d^4\,e}{5}-\frac {2\,B\,a^4\,d\,e^4}{5}-\frac {8\,A\,a\,b^3\,d^3\,e^2}{5}+\frac {8\,B\,a^3\,b\,d^2\,e^3}{5}+\frac {12\,A\,a^2\,b^2\,d^2\,e^3}{5}-\frac {12\,B\,a^2\,b^2\,d^3\,e^2}{5}-\frac {8\,A\,a^3\,b\,d\,e^4}{5}+\frac {8\,B\,a\,b^3\,d^4\,e}{5}}{e^6\,{\left (d+e\,x\right )}^{5/2}}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{e^6} \]

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)^2)/(d + e*x)^(7/2),x)

[Out]

((d + e*x)^(3/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(3*e^6) - ((d + e*x)*((2*B*a^4*e^4)/3 + (10*B*b^4*d^4
)/3 + (8*A*a^3*b*e^4)/3 - (8*A*b^4*d^3*e)/3 + 8*A*a*b^3*d^2*e^2 - 8*A*a^2*b^2*d*e^3 + 12*B*a^2*b^2*d^2*e^2 - (
32*B*a*b^3*d^3*e)/3 - (16*B*a^3*b*d*e^3)/3) + (d + e*x)^2*(8*B*a^3*b*e^3 - 20*B*b^4*d^3 + 12*A*b^4*d^2*e + 12*
A*a^2*b^2*e^3 - 36*B*a^2*b^2*d*e^2 - 24*A*a*b^3*d*e^2 + 48*B*a*b^3*d^2*e) + (2*A*a^4*e^5)/5 - (2*B*b^4*d^5)/5
+ (2*A*b^4*d^4*e)/5 - (2*B*a^4*d*e^4)/5 - (8*A*a*b^3*d^3*e^2)/5 + (8*B*a^3*b*d^2*e^3)/5 + (12*A*a^2*b^2*d^2*e^
3)/5 - (12*B*a^2*b^2*d^3*e^2)/5 - (8*A*a^3*b*d*e^4)/5 + (8*B*a*b^3*d^4*e)/5)/(e^6*(d + e*x)^(5/2)) + (2*B*b^4*
(d + e*x)^(5/2))/(5*e^6) + (4*b^2*(a*e - b*d)*(d + e*x)^(1/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/e^6

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sympy [A]  time = 5.28, size = 2440, normalized size = 11.40 \[ \text {result too large to display} \]

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)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**4*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 16*A*a**3*b*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) -
40*A*a**3*b*e**5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 96
*A*a**2*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) -
240*A*a**2*b**2*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 180*A*a**2*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) + 384*A*a*b**3*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) + 960*A*a*b**3*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(
d + e*x)) + 720*A*a*b**3*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sq
rt(d + e*x)) + 120*A*a*b**3*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*s
qrt(d + e*x)) - 256*A*b**4*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(
d + e*x)) - 640*A*b**4*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt
(d + e*x)) - 480*A*b**4*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*
sqrt(d + e*x)) - 80*A*b**4*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*
sqrt(d + e*x)) + 10*A*b**4*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sq
rt(d + e*x)) - 4*B*a**4*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) - 10*B*a**4*e**5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 64*B*a**3*b*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 160*B*a**3*b*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)
) - 120*B*a**3*b*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x
)) + 576*B*a**2*b**2*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d +
e*x)) + 1440*B*a**2*b**2*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*s
qrt(d + e*x)) + 1080*B*a**2*b**2*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8
*x**2*sqrt(d + e*x)) + 180*B*a**2*b**2*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*
e**8*x**2*sqrt(d + e*x)) - 1024*B*a*b**3*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e
**8*x**2*sqrt(d + e*x)) - 2560*B*a*b**3*d**3*e**2*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) +
15*e**8*x**2*sqrt(d + e*x)) - 1920*B*a*b**3*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d +
e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 320*B*a*b**3*d*e**4*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d
+ e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 40*B*a*b**3*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d
+ e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 512*B*b**4*d**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) + 1280*B*b**4*d**4*e*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x)
+ 15*e**8*x**2*sqrt(d + e*x)) + 960*B*b**4*d**3*e**2*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d +
e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 160*B*b**4*d**2*e**3*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(
d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*B*b**4*d*e**4*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(
d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 6*B*b**4*e**5*x**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d +
e*x) + 15*e**8*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**4*x + 2*A*a**3*b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b**3*x
**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*B*a**3*b*x**3/3 + 3*B*a**2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x*
*6/6)/d**(7/2), True))

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