3.18.98 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{5/2}} \, dx\) [1798]

Optimal. Leaf size=214 \[ \frac {2 (b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^{3/2}}-\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e)}{e^6 \sqrt {d+e x}}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) \sqrt {d+e x}}{e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{3/2}}{3 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{5/2}}{5 e^6}+\frac {2 b^4 B (d+e x)^{7/2}}{7 e^6} \]

[Out]

2/3*(-a*e+b*d)^4*(-A*e+B*d)/e^6/(e*x+d)^(3/2)+4/3*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*x+d)^(3/2)/e^6-
2/5*b^3*(-A*b*e-4*B*a*e+5*B*b*d)*(e*x+d)^(5/2)/e^6+2/7*b^4*B*(e*x+d)^(7/2)/e^6-2*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+
5*B*b*d)/e^6/(e*x+d)^(1/2)-4*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)*(e*x+d)^(1/2)/e^6

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Rubi [A]
time = 0.06, 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, 78} \begin {gather*} -\frac {2 b^3 (d+e x)^{5/2} (-4 a B e-A b e+5 b B d)}{5 e^6}+\frac {4 b^2 (d+e x)^{3/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6}-\frac {4 b \sqrt {d+e x} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6}-\frac {2 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^{3/2}}+\frac {2 b^4 B (d+e x)^{7/2}}{7 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.33, size = 336, normalized size = 1.57 \begin {gather*} \frac {-70 a^4 e^4 (2 B d+A e+3 B e x)+280 a^3 b e^3 \left (-A e (2 d+3 e x)+B \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+420 a^2 b^2 e^2 \left (A e \left (8 d^2+12 d e x+3 e^2 x^2\right )+B \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )\right )+56 a b^3 e \left (5 A e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+B \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )+2 b^4 \left (7 A e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )-5 B \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{105 e^6 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-70*a^4*e^4*(2*B*d + A*e + 3*B*e*x) + 280*a^3*b*e^3*(-(A*e*(2*d + 3*e*x)) + B*(8*d^2 + 12*d*e*x + 3*e^2*x^2))
+ 420*a^2*b^2*e^2*(A*e*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + B*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3)) + 5
6*a*b^3*e*(5*A*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + B*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 -
8*d*e^3*x^3 + 3*e^4*x^4)) + 2*b^4*(7*A*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) -
5*B*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(105*e^6*(d + e*x)^(
3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(504\) vs. \(2(194)=388\).
time = 0.90, size = 505, normalized size = 2.36 Too large to display

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)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/e^6*(1/7*B*b^4*(e*x+d)^(7/2)+1/5*A*b^4*e*(e*x+d)^(5/2)+4/5*B*a*b^3*e*(e*x+d)^(5/2)-B*b^4*d*(e*x+d)^(5/2)+4/3
*A*a*b^3*e^2*(e*x+d)^(3/2)-4/3*A*b^4*d*e*(e*x+d)^(3/2)+2*B*a^2*b^2*e^2*(e*x+d)^(3/2)-16/3*B*a*b^3*d*e*(e*x+d)^
(3/2)+10/3*B*b^4*d^2*(e*x+d)^(3/2)+6*A*a^2*b^2*e^3*(e*x+d)^(1/2)-12*A*a*b^3*d*e^2*(e*x+d)^(1/2)+6*A*b^4*d^2*e*
(e*x+d)^(1/2)+4*B*a^3*b*e^3*(e*x+d)^(1/2)-18*B*a^2*b^2*d*e^2*(e*x+d)^(1/2)+24*B*a*b^3*d^2*e*(e*x+d)^(1/2)-10*B
*b^4*d^3*(e*x+d)^(1/2)-1/3*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3-4*A*a*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*
d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/(e*x+d)^(3/2)-(4*A*a^3*b*e^4-12*A*a^2*b
^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+B*a^4*e^4-8*B*a^3*b*d*e^3+18*B*a^2*b^2*d^2*e^2-16*B*a*b^3*d^3*e+5*B*
b^4*d^4)/(e*x+d)^(1/2))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (207) = 414\).
time = 0.29, size = 432, normalized size = 2.02 \begin {gather*} \frac {2}{105} \, {\left ({\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} - 21 \, {\left (5 \, B b^{4} d - 4 \, B a b^{3} e - A b^{4} e\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 70 \, {\left (5 \, B b^{4} d^{2} + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2} - 2 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d\right )} {\left (x e + d\right )}^{\frac {3}{2}} - 210 \, {\left (5 \, B b^{4} d^{3} - 2 \, B a^{3} b e^{3} - 3 \, A a^{2} b^{2} e^{3} - 3 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{2} + 3 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d\right )} \sqrt {x e + d}\right )} e^{\left (-5\right )} + \frac {35 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{4} + 2 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d^{3} - 2 \, {\left (2 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} d^{2} - 3 \, {\left (5 \, B b^{4} d^{4} + B a^{4} e^{4} + 4 \, A a^{3} b e^{4} - 4 \, {\left (4 \, B a b^{3} e + A b^{4} e\right )} d^{3} + 6 \, {\left (3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} d^{2} - 4 \, {\left (2 \, B a^{3} b e^{3} + 3 \, A a^{2} b^{2} e^{3}\right )} d\right )} {\left (x e + d\right )} + {\left (B a^{4} e^{4} + 4 \, A a^{3} b e^{4}\right )} d\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{\frac {3}{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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/105*((15*(x*e + d)^(7/2)*B*b^4 - 21*(5*B*b^4*d - 4*B*a*b^3*e - A*b^4*e)*(x*e + d)^(5/2) + 70*(5*B*b^4*d^2 +
3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2 - 2*(4*B*a*b^3*e + A*b^4*e)*d)*(x*e + d)^(3/2) - 210*(5*B*b^4*d^3 - 2*B*a^3*b*
e^3 - 3*A*a^2*b^2*e^3 - 3*(4*B*a*b^3*e + A*b^4*e)*d^2 + 3*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d)*sqrt(x*e + d))*
e^(-5) + 35*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3*e + A*b^4*e)*d^4 + 2*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d^3 - 2
*(2*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*d^2 - 3*(5*B*b^4*d^4 + B*a^4*e^4 + 4*A*a^3*b*e^4 - 4*(4*B*a*b^3*e + A*b^4*e
)*d^3 + 6*(3*B*a^2*b^2*e^2 + 2*A*a*b^3*e^2)*d^2 - 4*(2*B*a^3*b*e^3 + 3*A*a^2*b^2*e^3)*d)*(x*e + d) + (B*a^4*e^
4 + 4*A*a^3*b*e^4)*d)*e^(-5)/(x*e + d)^(3/2))*e^(-1)

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Fricas [A]
time = 1.42, size = 407, normalized size = 1.90 \begin {gather*} -\frac {2 \, {\left (1280 \, B b^{4} d^{5} - {\left (15 \, B b^{4} x^{5} - 35 \, A a^{4} + 21 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 70 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 105 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} e^{5} + 2 \, {\left (15 \, B b^{4} d x^{4} + 28 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d x^{3} + 210 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d x^{2} - 420 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d x + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d\right )} e^{4} - 16 \, {\left (5 \, B b^{4} d^{2} x^{3} + 21 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} x^{2} - 105 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} x + 35 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2}\right )} e^{3} + 32 \, {\left (15 \, B b^{4} d^{3} x^{2} - 42 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} x + 35 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3}\right )} e^{2} + 128 \, {\left (15 \, B b^{4} d^{4} x - 7 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4}\right )} e\right )} \sqrt {x e + d}}{105 \, {\left (x^{2} e^{8} + 2 \, d x e^{7} + d^{2} e^{6}\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)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(1280*B*b^4*d^5 - (15*B*b^4*x^5 - 35*A*a^4 + 21*(4*B*a*b^3 + A*b^4)*x^4 + 70*(3*B*a^2*b^2 + 2*A*a*b^3)*
x^3 + 210*(2*B*a^3*b + 3*A*a^2*b^2)*x^2 - 105*(B*a^4 + 4*A*a^3*b)*x)*e^5 + 2*(15*B*b^4*d*x^4 + 28*(4*B*a*b^3 +
A*b^4)*d*x^3 + 210*(3*B*a^2*b^2 + 2*A*a*b^3)*d*x^2 - 420*(2*B*a^3*b + 3*A*a^2*b^2)*d*x + 35*(B*a^4 + 4*A*a^3*
b)*d)*e^4 - 16*(5*B*b^4*d^2*x^3 + 21*(4*B*a*b^3 + A*b^4)*d^2*x^2 - 105*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*x + 35*(2
*B*a^3*b + 3*A*a^2*b^2)*d^2)*e^3 + 32*(15*B*b^4*d^3*x^2 - 42*(4*B*a*b^3 + A*b^4)*d^3*x + 35*(3*B*a^2*b^2 + 2*A
*a*b^3)*d^3)*e^2 + 128*(15*B*b^4*d^4*x - 7*(4*B*a*b^3 + A*b^4)*d^4)*e)*sqrt(x*e + d)/(x^2*e^8 + 2*d*x*e^7 + d^
2*e^6)

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Sympy [A]
time = 48.19, size = 304, normalized size = 1.42 \begin {gather*} \frac {2 B b^{4} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A b^{4} e + 8 B a b^{3} e - 10 B b^{4} d\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (8 A a b^{3} e^{2} - 8 A b^{4} d e + 12 B a^{2} b^{2} e^{2} - 32 B a b^{3} d e + 20 B b^{4} d^{2}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (12 A a^{2} b^{2} e^{3} - 24 A a b^{3} d e^{2} + 12 A b^{4} d^{2} e + 8 B a^{3} b e^{3} - 36 B a^{2} b^{2} d e^{2} + 48 B a b^{3} d^{2} e - 20 B b^{4} d^{3}\right )}{e^{6}} - \frac {2 \left (a e - b d\right )^{3} \cdot \left (4 A b e + B a e - 5 B b d\right )}{e^{6} \sqrt {d + e x}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{4}}{3 e^{6} \left (d + e x\right )^{\frac {3}{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)**2/(e*x+d)**(5/2),x)

[Out]

2*B*b**4*(d + e*x)**(7/2)/(7*e**6) + (d + e*x)**(5/2)*(2*A*b**4*e + 8*B*a*b**3*e - 10*B*b**4*d)/(5*e**6) + (d
+ e*x)**(3/2)*(8*A*a*b**3*e**2 - 8*A*b**4*d*e + 12*B*a**2*b**2*e**2 - 32*B*a*b**3*d*e + 20*B*b**4*d**2)/(3*e**
6) + sqrt(d + e*x)*(12*A*a**2*b**2*e**3 - 24*A*a*b**3*d*e**2 + 12*A*b**4*d**2*e + 8*B*a**3*b*e**3 - 36*B*a**2*
b**2*d*e**2 + 48*B*a*b**3*d**2*e - 20*B*b**4*d**3)/e**6 - 2*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b*d)/(e**6*s
qrt(d + e*x)) + 2*(-A*e + B*d)*(a*e - b*d)**4/(3*e**6*(d + e*x)**(3/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (207) = 414\).
time = 0.90, size = 567, normalized size = 2.65 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} e^{36} - 105 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d e^{36} + 350 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{2} e^{36} - 1050 \, \sqrt {x e + d} B b^{4} d^{3} e^{36} + 84 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} e^{37} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} e^{37} - 560 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d e^{37} - 140 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d e^{37} + 2520 \, \sqrt {x e + d} B a b^{3} d^{2} e^{37} + 630 \, \sqrt {x e + d} A b^{4} d^{2} e^{37} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} e^{38} + 140 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} e^{38} - 1890 \, \sqrt {x e + d} B a^{2} b^{2} d e^{38} - 1260 \, \sqrt {x e + d} A a b^{3} d e^{38} + 420 \, \sqrt {x e + d} B a^{3} b e^{39} + 630 \, \sqrt {x e + d} A a^{2} b^{2} e^{39}\right )} e^{\left (-42\right )} - \frac {2 \, {\left (15 \, {\left (x e + d\right )} B b^{4} d^{4} - B b^{4} d^{5} - 48 \, {\left (x e + d\right )} B a b^{3} d^{3} e - 12 \, {\left (x e + d\right )} A b^{4} d^{3} e + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 54 \, {\left (x e + d\right )} B a^{2} b^{2} d^{2} e^{2} + 36 \, {\left (x e + d\right )} A a b^{3} d^{2} e^{2} - 6 \, B a^{2} b^{2} d^{3} e^{2} - 4 \, A a b^{3} d^{3} e^{2} - 24 \, {\left (x e + d\right )} B a^{3} b d e^{3} - 36 \, {\left (x e + d\right )} A a^{2} b^{2} d e^{3} + 4 \, B a^{3} b d^{2} e^{3} + 6 \, A a^{2} b^{2} d^{2} e^{3} + 3 \, {\left (x e + d\right )} B a^{4} e^{4} + 12 \, {\left (x e + d\right )} A a^{3} b e^{4} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} + A a^{4} e^{5}\right )} e^{\left (-6\right )}}{3 \, {\left (x e + d\right )}^{\frac {3}{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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/105*(15*(x*e + d)^(7/2)*B*b^4*e^36 - 105*(x*e + d)^(5/2)*B*b^4*d*e^36 + 350*(x*e + d)^(3/2)*B*b^4*d^2*e^36 -
1050*sqrt(x*e + d)*B*b^4*d^3*e^36 + 84*(x*e + d)^(5/2)*B*a*b^3*e^37 + 21*(x*e + d)^(5/2)*A*b^4*e^37 - 560*(x*
e + d)^(3/2)*B*a*b^3*d*e^37 - 140*(x*e + d)^(3/2)*A*b^4*d*e^37 + 2520*sqrt(x*e + d)*B*a*b^3*d^2*e^37 + 630*sqr
t(x*e + d)*A*b^4*d^2*e^37 + 210*(x*e + d)^(3/2)*B*a^2*b^2*e^38 + 140*(x*e + d)^(3/2)*A*a*b^3*e^38 - 1890*sqrt(
x*e + d)*B*a^2*b^2*d*e^38 - 1260*sqrt(x*e + d)*A*a*b^3*d*e^38 + 420*sqrt(x*e + d)*B*a^3*b*e^39 + 630*sqrt(x*e
+ d)*A*a^2*b^2*e^39)*e^(-42) - 2/3*(15*(x*e + d)*B*b^4*d^4 - B*b^4*d^5 - 48*(x*e + d)*B*a*b^3*d^3*e - 12*(x*e
+ d)*A*b^4*d^3*e + 4*B*a*b^3*d^4*e + A*b^4*d^4*e + 54*(x*e + d)*B*a^2*b^2*d^2*e^2 + 36*(x*e + d)*A*a*b^3*d^2*e
^2 - 6*B*a^2*b^2*d^3*e^2 - 4*A*a*b^3*d^3*e^2 - 24*(x*e + d)*B*a^3*b*d*e^3 - 36*(x*e + d)*A*a^2*b^2*d*e^3 + 4*B
*a^3*b*d^2*e^3 + 6*A*a^2*b^2*d^2*e^3 + 3*(x*e + d)*B*a^4*e^4 + 12*(x*e + d)*A*a^3*b*e^4 - B*a^4*d*e^4 - 4*A*a^
3*b*d*e^4 + A*a^4*e^5)*e^(-6)/(x*e + d)^(3/2)

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Mupad [B]
time = 1.94, size = 367, normalized size = 1.71 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (2\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3+8\,A\,a^3\,b\,e^4+36\,B\,a^2\,b^2\,d^2\,e^2-24\,A\,a^2\,b^2\,d\,e^3-32\,B\,a\,b^3\,d^3\,e+24\,A\,a\,b^3\,d^2\,e^2+10\,B\,b^4\,d^4-8\,A\,b^4\,d^3\,e\right )+\frac {2\,A\,a^4\,e^5}{3}-\frac {2\,B\,b^4\,d^5}{3}+\frac {2\,A\,b^4\,d^4\,e}{3}-\frac {2\,B\,a^4\,d\,e^4}{3}-\frac {8\,A\,a\,b^3\,d^3\,e^2}{3}+\frac {8\,B\,a^3\,b\,d^2\,e^3}{3}+4\,A\,a^2\,b^2\,d^2\,e^3-4\,B\,a^2\,b^2\,d^3\,e^2-\frac {8\,A\,a^3\,b\,d\,e^4}{3}+\frac {8\,B\,a\,b^3\,d^4\,e}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{3\,e^6} \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)^2)/(d + e*x)^(5/2),x)

[Out]

((d + e*x)^(5/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(5*e^6) - ((d + e*x)*(2*B*a^4*e^4 + 10*B*b^4*d^4 + 8*
A*a^3*b*e^4 - 8*A*b^4*d^3*e + 24*A*a*b^3*d^2*e^2 - 24*A*a^2*b^2*d*e^3 + 36*B*a^2*b^2*d^2*e^2 - 32*B*a*b^3*d^3*
e - 16*B*a^3*b*d*e^3) + (2*A*a^4*e^5)/3 - (2*B*b^4*d^5)/3 + (2*A*b^4*d^4*e)/3 - (2*B*a^4*d*e^4)/3 - (8*A*a*b^3
*d^3*e^2)/3 + (8*B*a^3*b*d^2*e^3)/3 + 4*A*a^2*b^2*d^2*e^3 - 4*B*a^2*b^2*d^3*e^2 - (8*A*a^3*b*d*e^4)/3 + (8*B*a
*b^3*d^4*e)/3)/(e^6*(d + e*x)^(3/2)) + (2*B*b^4*(d + e*x)^(7/2))/(7*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(1/2)*
(3*A*b*e + 2*B*a*e - 5*B*b*d))/e^6 + (4*b^2*(a*e - b*d)*(d + e*x)^(3/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(3*e^6)

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