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

Optimal. Leaf size=214 \[ -\frac {2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac {4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac {4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac {2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt {d+e x}}+\frac {2 b^4 B (d+e x)^{9/2}}{9 e^6} \]

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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} \begin {gather*} -\frac {2 b^3 (d+e x)^{7/2} (-4 a B e-A b e+5 b B d)}{7 e^6}+\frac {4 b^2 (d+e x)^{5/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{5 e^6}-\frac {4 b (d+e x)^{3/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{3 e^6}+\frac {2 \sqrt {d+e x} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6}+\frac {2 (b d-a e)^4 (B d-A e)}{e^6 \sqrt {d+e x}}+\frac {2 b^4 B (d+e x)^{9/2}}{9 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)^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 183, normalized size = 0.86 \begin {gather*} \frac {2 \left (-45 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+126 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-210 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+315 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)+315 (b d-a e)^4 (B d-A e)+35 b^4 B (d+e x)^5\right )}{315 e^6 \sqrt {d+e x}} \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)^(3/2),x]

[Out]

(2*(315*(b*d - a*e)^4*(B*d - A*e) + 315*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x) - 210*b*(b*d - a*e
)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 126*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^3
- 45*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 35*b^4*B*(d + e*x)^5))/(315*e^6*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.23, size = 543, normalized size = 2.54 \begin {gather*} \frac {2 \left (-315 a^4 A e^5+315 a^4 B e^4 (d+e x)+315 a^4 B d e^4+1260 a^3 A b e^4 (d+e x)+1260 a^3 A b d e^4-1260 a^3 b B d^2 e^3-2520 a^3 b B d e^3 (d+e x)+420 a^3 b B e^3 (d+e x)^2-1890 a^2 A b^2 d^2 e^3-3780 a^2 A b^2 d e^3 (d+e x)+630 a^2 A b^2 e^3 (d+e x)^2+1890 a^2 b^2 B d^3 e^2+5670 a^2 b^2 B d^2 e^2 (d+e x)-1890 a^2 b^2 B d e^2 (d+e x)^2+378 a^2 b^2 B e^2 (d+e x)^3+1260 a A b^3 d^3 e^2+3780 a A b^3 d^2 e^2 (d+e x)-1260 a A b^3 d e^2 (d+e x)^2+252 a A b^3 e^2 (d+e x)^3-1260 a b^3 B d^4 e-5040 a b^3 B d^3 e (d+e x)+2520 a b^3 B d^2 e (d+e x)^2-1008 a b^3 B d e (d+e x)^3+180 a b^3 B e (d+e x)^4-315 A b^4 d^4 e-1260 A b^4 d^3 e (d+e x)+630 A b^4 d^2 e (d+e x)^2-252 A b^4 d e (d+e x)^3+45 A b^4 e (d+e x)^4+315 b^4 B d^5+1575 b^4 B d^4 (d+e x)-1050 b^4 B d^3 (d+e x)^2+630 b^4 B d^2 (d+e x)^3-225 b^4 B d (d+e x)^4+35 b^4 B (d+e x)^5\right )}{315 e^6 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(315*b^4*B*d^5 - 315*A*b^4*d^4*e - 1260*a*b^3*B*d^4*e + 1260*a*A*b^3*d^3*e^2 + 1890*a^2*b^2*B*d^3*e^2 - 189
0*a^2*A*b^2*d^2*e^3 - 1260*a^3*b*B*d^2*e^3 + 1260*a^3*A*b*d*e^4 + 315*a^4*B*d*e^4 - 315*a^4*A*e^5 + 1575*b^4*B
*d^4*(d + e*x) - 1260*A*b^4*d^3*e*(d + e*x) - 5040*a*b^3*B*d^3*e*(d + e*x) + 3780*a*A*b^3*d^2*e^2*(d + e*x) +
5670*a^2*b^2*B*d^2*e^2*(d + e*x) - 3780*a^2*A*b^2*d*e^3*(d + e*x) - 2520*a^3*b*B*d*e^3*(d + e*x) + 1260*a^3*A*
b*e^4*(d + e*x) + 315*a^4*B*e^4*(d + e*x) - 1050*b^4*B*d^3*(d + e*x)^2 + 630*A*b^4*d^2*e*(d + e*x)^2 + 2520*a*
b^3*B*d^2*e*(d + e*x)^2 - 1260*a*A*b^3*d*e^2*(d + e*x)^2 - 1890*a^2*b^2*B*d*e^2*(d + e*x)^2 + 630*a^2*A*b^2*e^
3*(d + e*x)^2 + 420*a^3*b*B*e^3*(d + e*x)^2 + 630*b^4*B*d^2*(d + e*x)^3 - 252*A*b^4*d*e*(d + e*x)^3 - 1008*a*b
^3*B*d*e*(d + e*x)^3 + 252*a*A*b^3*e^2*(d + e*x)^3 + 378*a^2*b^2*B*e^2*(d + e*x)^3 - 225*b^4*B*d*(d + e*x)^4 +
45*A*b^4*e*(d + e*x)^4 + 180*a*b^3*B*e*(d + e*x)^4 + 35*b^4*B*(d + e*x)^5))/(315*e^6*Sqrt[d + e*x])

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fricas [B]  time = 0.41, size = 418, normalized size = 1.95 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} e^{5} x^{5} + 1280 \, B b^{4} d^{5} - 315 \, A a^{4} e^{5} - 1152 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2016 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 1680 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 630 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 5 \, {\left (10 \, B b^{4} d e^{4} - 9 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \, {\left (40 \, B b^{4} d^{2} e^{3} - 36 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 63 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 2 \, {\left (80 \, B b^{4} d^{3} e^{2} - 72 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 126 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 105 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + {\left (640 \, B b^{4} d^{4} e - 576 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1008 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 840 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 315 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{7} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^4*e^5*x^5 + 1280*B*b^4*d^5 - 315*A*a^4*e^5 - 1152*(4*B*a*b^3 + A*b^4)*d^4*e + 2016*(3*B*a^2*b^2
+ 2*A*a*b^3)*d^3*e^2 - 1680*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 630*(B*a^4 + 4*A*a^3*b)*d*e^4 - 5*(10*B*b^4*d*
e^4 - 9*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 2*(40*B*b^4*d^2*e^3 - 36*(4*B*a*b^3 + A*b^4)*d*e^4 + 63*(3*B*a^2*b^2 +
2*A*a*b^3)*e^5)*x^3 - 2*(80*B*b^4*d^3*e^2 - 72*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 126*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e
^4 - 105*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + (640*B*b^4*d^4*e - 576*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 1008*(3*B*a
^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 840*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 315*(B*a^4 + 4*A*a^3*b)*e^5)*x)*sqrt(e*x +
d)/(e^7*x + d*e^6)

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giac [B]  time = 0.25, size = 587, normalized size = 2.74 \begin {gather*} \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{4} e^{48} - 225 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{48} + 630 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{48} - 1050 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{48} + 1575 \, \sqrt {x e + d} B b^{4} d^{4} e^{48} + 180 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{49} + 45 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{49} - 1008 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{49} - 252 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{49} + 2520 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{49} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{49} - 5040 \, \sqrt {x e + d} B a b^{3} d^{3} e^{49} - 1260 \, \sqrt {x e + d} A b^{4} d^{3} e^{49} + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{50} + 252 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{50} - 1890 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{50} - 1260 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{50} + 5670 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{50} + 3780 \, \sqrt {x e + d} A a b^{3} d^{2} e^{50} + 420 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{51} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{51} - 2520 \, \sqrt {x e + d} B a^{3} b d e^{51} - 3780 \, \sqrt {x e + d} A a^{2} b^{2} d e^{51} + 315 \, \sqrt {x e + d} B a^{4} e^{52} + 1260 \, \sqrt {x e + d} A a^{3} b e^{52}\right )} e^{\left (-54\right )} + \frac {2 \, {\left (B b^{4} d^{5} - 4 \, B a b^{3} d^{4} e - A b^{4} d^{4} e + 6 \, B a^{2} b^{2} d^{3} e^{2} + 4 \, A a b^{3} d^{3} e^{2} - 4 \, B a^{3} b d^{2} e^{3} - 6 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} - A a^{4} e^{5}\right )} e^{\left (-6\right )}}{\sqrt {x e + d}} \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)^(3/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*B*b^4*e^48 - 225*(x*e + d)^(7/2)*B*b^4*d*e^48 + 630*(x*e + d)^(5/2)*B*b^4*d^2*e^48 -
1050*(x*e + d)^(3/2)*B*b^4*d^3*e^48 + 1575*sqrt(x*e + d)*B*b^4*d^4*e^48 + 180*(x*e + d)^(7/2)*B*a*b^3*e^49 +
45*(x*e + d)^(7/2)*A*b^4*e^49 - 1008*(x*e + d)^(5/2)*B*a*b^3*d*e^49 - 252*(x*e + d)^(5/2)*A*b^4*d*e^49 + 2520*
(x*e + d)^(3/2)*B*a*b^3*d^2*e^49 + 630*(x*e + d)^(3/2)*A*b^4*d^2*e^49 - 5040*sqrt(x*e + d)*B*a*b^3*d^3*e^49 -
1260*sqrt(x*e + d)*A*b^4*d^3*e^49 + 378*(x*e + d)^(5/2)*B*a^2*b^2*e^50 + 252*(x*e + d)^(5/2)*A*a*b^3*e^50 - 18
90*(x*e + d)^(3/2)*B*a^2*b^2*d*e^50 - 1260*(x*e + d)^(3/2)*A*a*b^3*d*e^50 + 5670*sqrt(x*e + d)*B*a^2*b^2*d^2*e
^50 + 3780*sqrt(x*e + d)*A*a*b^3*d^2*e^50 + 420*(x*e + d)^(3/2)*B*a^3*b*e^51 + 630*(x*e + d)^(3/2)*A*a^2*b^2*e
^51 - 2520*sqrt(x*e + d)*B*a^3*b*d*e^51 - 3780*sqrt(x*e + d)*A*a^2*b^2*d*e^51 + 315*sqrt(x*e + d)*B*a^4*e^52 +
1260*sqrt(x*e + d)*A*a^3*b*e^52)*e^(-54) + 2*(B*b^4*d^5 - 4*B*a*b^3*d^4*e - A*b^4*d^4*e + 6*B*a^2*b^2*d^3*e^2
+ 4*A*a*b^3*d^3*e^2 - 4*B*a^3*b*d^2*e^3 - 6*A*a^2*b^2*d^2*e^3 + B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 - A*a^4*e^5)*e^
(-6)/sqrt(x*e + d)

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maple [B]  time = 0.07, size = 469, normalized size = 2.19 \begin {gather*} -\frac {2 \left (-35 b^{4} B \,x^{5} e^{5}-45 A \,b^{4} e^{5} x^{4}-180 B a \,b^{3} e^{5} x^{4}+50 B \,b^{4} d \,e^{4} x^{4}-252 A a \,b^{3} e^{5} x^{3}+72 A \,b^{4} d \,e^{4} x^{3}-378 B \,a^{2} b^{2} e^{5} x^{3}+288 B a \,b^{3} d \,e^{4} x^{3}-80 B \,b^{4} d^{2} e^{3} x^{3}-630 A \,a^{2} b^{2} e^{5} x^{2}+504 A a \,b^{3} d \,e^{4} x^{2}-144 A \,b^{4} d^{2} e^{3} x^{2}-420 B \,a^{3} b \,e^{5} x^{2}+756 B \,a^{2} b^{2} d \,e^{4} x^{2}-576 B a \,b^{3} d^{2} e^{3} x^{2}+160 B \,b^{4} d^{3} e^{2} x^{2}-1260 A \,a^{3} b \,e^{5} x +2520 A \,a^{2} b^{2} d \,e^{4} x -2016 A a \,b^{3} d^{2} e^{3} x +576 A \,b^{4} d^{3} e^{2} x -315 B \,a^{4} e^{5} x +1680 B \,a^{3} b d \,e^{4} x -3024 B \,a^{2} b^{2} d^{2} e^{3} x +2304 B a \,b^{3} d^{3} e^{2} x -640 B \,b^{4} d^{4} e x +315 A \,a^{4} e^{5}-2520 A \,a^{3} b d \,e^{4}+5040 A \,a^{2} b^{2} d^{2} e^{3}-4032 A a \,b^{3} d^{3} e^{2}+1152 A \,b^{4} d^{4} e -630 B \,a^{4} d \,e^{4}+3360 B \,d^{2} a^{3} b \,e^{3}-6048 B \,d^{3} a^{2} b^{2} e^{2}+4608 B a \,b^{3} d^{4} e -1280 B \,b^{4} d^{5}\right )}{315 \sqrt {e x +d}\, e^{6}} \end {gather*}

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)^(3/2),x)

[Out]

-2/315*(-35*B*b^4*e^5*x^5-45*A*b^4*e^5*x^4-180*B*a*b^3*e^5*x^4+50*B*b^4*d*e^4*x^4-252*A*a*b^3*e^5*x^3+72*A*b^4
*d*e^4*x^3-378*B*a^2*b^2*e^5*x^3+288*B*a*b^3*d*e^4*x^3-80*B*b^4*d^2*e^3*x^3-630*A*a^2*b^2*e^5*x^2+504*A*a*b^3*
d*e^4*x^2-144*A*b^4*d^2*e^3*x^2-420*B*a^3*b*e^5*x^2+756*B*a^2*b^2*d*e^4*x^2-576*B*a*b^3*d^2*e^3*x^2+160*B*b^4*
d^3*e^2*x^2-1260*A*a^3*b*e^5*x+2520*A*a^2*b^2*d*e^4*x-2016*A*a*b^3*d^2*e^3*x+576*A*b^4*d^3*e^2*x-315*B*a^4*e^5
*x+1680*B*a^3*b*d*e^4*x-3024*B*a^2*b^2*d^2*e^3*x+2304*B*a*b^3*d^3*e^2*x-640*B*b^4*d^4*e*x+315*A*a^4*e^5-2520*A
*a^3*b*d*e^4+5040*A*a^2*b^2*d^2*e^3-4032*A*a*b^3*d^3*e^2+1152*A*b^4*d^4*e-630*B*a^4*d*e^4+3360*B*a^3*b*d^2*e^3
-6048*B*a^2*b^2*d^3*e^2+4608*B*a*b^3*d^4*e-1280*B*b^4*d^5)/(e*x+d)^(1/2)/e^6

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maxima [B]  time = 0.62, size = 417, normalized size = 1.95 \begin {gather*} \frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{4} - 45 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 126 \, {\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 )} {\left (e x + d\right )}^{\frac {5}{2}} - 210 \, {\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 )}^{\frac {3}{2}} + 315 \, {\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 )} \sqrt {e x + d}}{e^{5}} + \frac {315 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}}{\sqrt {e x + d} e^{5}}\right )}}{315 \, e} \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)^(3/2),x, algorithm="maxima")

[Out]

2/315*((35*(e*x + d)^(9/2)*B*b^4 - 45*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(7/2) + 126*(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)*(e*x + d)^(5/2) - 210*(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)^(3/2) + 315*(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)*sqrt(e*x + d))/e^5 + 315*(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*
(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d*e^4)/(sqrt(e*x
+ d)*e^5))/e

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mupad [B]  time = 1.95, size = 296, normalized size = 1.38 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{7\,e^6}-\frac {-2\,B\,a^4\,d\,e^4+2\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3-8\,A\,a^3\,b\,d\,e^4-12\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+8\,B\,a\,b^3\,d^4\,e-8\,A\,a\,b^3\,d^3\,e^2-2\,B\,b^4\,d^5+2\,A\,b^4\,d^4\,e}{e^6\,\sqrt {d+e\,x}}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{3\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{5\,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)^(3/2),x)

[Out]

((d + e*x)^(7/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(7*e^6) - (2*A*a^4*e^5 - 2*B*b^4*d^5 + 2*A*b^4*d^4*e
- 2*B*a^4*d*e^4 - 8*A*a*b^3*d^3*e^2 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3 - 12*B*a^2*b^2*d^3*e^2 - 8*A*a^
3*b*d*e^4 + 8*B*a*b^3*d^4*e)/(e^6*(d + e*x)^(1/2)) + (2*(a*e - b*d)^3*(d + e*x)^(1/2)*(4*A*b*e + B*a*e - 5*B*b
*d))/e^6 + (2*B*b^4*(d + e*x)^(9/2))/(9*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(3/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d
))/(3*e^6) + (4*b^2*(a*e - b*d)*(d + e*x)^(5/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(5*e^6)

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sympy [A]  time = 120.61, size = 394, normalized size = 1.84 \begin {gather*} \frac {2 B b^{4} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (2 A b^{4} e + 8 B a b^{3} e - 10 B b^{4} d\right )}{7 e^{6}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \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 )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \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 )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (8 A a^{3} b e^{4} - 24 A a^{2} b^{2} d e^{3} + 24 A a b^{3} d^{2} e^{2} - 8 A b^{4} d^{3} e + 2 B a^{4} e^{4} - 16 B a^{3} b d e^{3} + 36 B a^{2} b^{2} d^{2} e^{2} - 32 B a b^{3} d^{3} e + 10 B b^{4} d^{4}\right )}{e^{6}} + \frac {2 \left (- A e + B d\right ) \left (a e - b d\right )^{4}}{e^{6} \sqrt {d + e x}} \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)**(3/2),x)

[Out]

2*B*b**4*(d + e*x)**(9/2)/(9*e**6) + (d + e*x)**(7/2)*(2*A*b**4*e + 8*B*a*b**3*e - 10*B*b**4*d)/(7*e**6) + (d
+ e*x)**(5/2)*(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)/(5*e**
6) + (d + e*x)**(3/2)*(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)/(3*e**6) + sqrt(d + e*x)*(8*A*a**3*b*e**4 - 24*A*a**2*b*
*2*d*e**3 + 24*A*a*b**3*d**2*e**2 - 8*A*b**4*d**3*e + 2*B*a**4*e**4 - 16*B*a**3*b*d*e**3 + 36*B*a**2*b**2*d**2
*e**2 - 32*B*a*b**3*d**3*e + 10*B*b**4*d**4)/e**6 + 2*(-A*e + B*d)*(a*e - b*d)**4/(e**6*sqrt(d + e*x))

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