3.1796 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=216 \[ -\frac {2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac {4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac {4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac {2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac {2 \sqrt {d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac {2 b^4 B (d+e x)^{11/2}}{11 e^6} \]

[Out]

2/3*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)*(e*x+d)^(3/2)/e^6-4/5*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)*(e*x
+d)^(5/2)/e^6+4/7*b^2*(-a*e+b*d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*(e*x+d)^(7/2)/e^6-2/9*b^3*(-A*b*e-4*B*a*e+5*B*b*d)
*(e*x+d)^(9/2)/e^6+2/11*b^4*B*(e*x+d)^(11/2)/e^6-2*(-a*e+b*d)^4*(-A*e+B*d)*(e*x+d)^(1/2)/e^6

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Rubi [A]  time = 0.09, antiderivative size = 216, 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} \[ -\frac {2 b^3 (d+e x)^{9/2} (-4 a B e-A b e+5 b B d)}{9 e^6}+\frac {4 b^2 (d+e x)^{7/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{7 e^6}-\frac {4 b (d+e x)^{5/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{5 e^6}+\frac {2 (d+e x)^{3/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{3 e^6}-\frac {2 \sqrt {d+e x} (b d-a e)^4 (B d-A e)}{e^6}+\frac {2 b^4 B (d+e x)^{11/2}}{11 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^4*(B*d - A*e)*Sqrt[d + e*x])/e^6 + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)^(3/2
))/(3*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(5/2))/(5*e^6) + (4*b^2*(b*d - a*e)*(5
*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(7*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^(9/2))/(9*
e^6) + (2*b^4*B*(d + e*x)^(11/2))/(11*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}{\sqrt {d+e x}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e)}{e^5 \sqrt {d+e x}}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) \sqrt {d+e x}}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^{3/2}}{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)^{5/2}}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{7/2}}{e^5}+\frac {b^4 B (d+e x)^{9/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4 (B d-A e) \sqrt {d+e x}}{e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{3/2}}{3 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)^{5/2}}{5 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)^{7/2}}{7 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{9/2}}{9 e^6}+\frac {2 b^4 B (d+e x)^{11/2}}{11 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 183, normalized size = 0.85 \[ \frac {2 \sqrt {d+e x} \left (-385 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+990 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-1386 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+1155 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-3465 (b d-a e)^4 (B d-A e)+315 b^4 B (d+e x)^5\right )}{3465 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-3465*(b*d - a*e)^4*(B*d - A*e) + 1155*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x) -
1386*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 990*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a
*B*e)*(d + e*x)^3 - 385*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 315*b^4*B*(d + e*x)^5))/(3465*e^6)

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fricas [B]  time = 0.92, size = 408, normalized size = 1.89 \[ \frac {2 \, {\left (315 \, B b^{4} e^{5} x^{5} - 1280 \, B b^{4} d^{5} + 3465 \, A a^{4} e^{5} + 1408 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 3168 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 3696 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 2310 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 35 \, {\left (10 \, B b^{4} d e^{4} - 11 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 10 \, {\left (40 \, B b^{4} d^{2} e^{3} - 44 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 99 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \, {\left (80 \, B b^{4} d^{3} e^{2} - 88 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 198 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 231 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + {\left (640 \, B b^{4} d^{4} e - 704 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 1584 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 1848 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 1155 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \]

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)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^4*e^5*x^5 - 1280*B*b^4*d^5 + 3465*A*a^4*e^5 + 1408*(4*B*a*b^3 + A*b^4)*d^4*e - 3168*(3*B*a^2*b
^2 + 2*A*a*b^3)*d^3*e^2 + 3696*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - 2310*(B*a^4 + 4*A*a^3*b)*d*e^4 - 35*(10*B*b
^4*d*e^4 - 11*(4*B*a*b^3 + A*b^4)*e^5)*x^4 + 10*(40*B*b^4*d^2*e^3 - 44*(4*B*a*b^3 + A*b^4)*d*e^4 + 99*(3*B*a^2
*b^2 + 2*A*a*b^3)*e^5)*x^3 - 6*(80*B*b^4*d^3*e^2 - 88*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 198*(3*B*a^2*b^2 + 2*A*a*b
^3)*d*e^4 - 231*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + (640*B*b^4*d^4*e - 704*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 1584
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 1848*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 1155*(B*a^4 + 4*A*a^3*b)*e^5)*x)*s
qrt(e*x + d)/e^6

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giac [B]  time = 0.20, size = 503, normalized size = 2.33 \[ \frac {2}{3465} \, {\left (1155 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a^{4} e^{\left (-1\right )} + 4620 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a^{3} b e^{\left (-1\right )} + 924 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} B a^{3} b e^{\left (-2\right )} + 1386 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} A a^{2} b^{2} e^{\left (-2\right )} + 594 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B a^{2} b^{2} e^{\left (-3\right )} + 396 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A a b^{3} e^{\left (-3\right )} + 44 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B a b^{3} e^{\left (-4\right )} + 11 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} A b^{4} e^{\left (-4\right )} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} B b^{4} e^{\left (-5\right )} + 3465 \, \sqrt {x e + d} A a^{4}\right )} e^{\left (-1\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)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^4*e^(-1) + 4620*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A
*a^3*b*e^(-1) + 924*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*b*e^(-2) + 1386*(3
*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b^2*e^(-2) + 594*(5*(x*e + d)^(7/2) - 21
*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^2*b^2*e^(-3) + 396*(5*(x*e + d)^(7/2)
- 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a*b^3*e^(-3) + 44*(35*(x*e + d)^(9/2
) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a*b^3
*e^(-4) + 11*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*A*b^4*e^(-4) + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^
2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*b^4*e^(-5) + 3465*sqrt(x*e
+ d)*A*a^4)*e^(-1)

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maple [B]  time = 0.05, size = 469, normalized size = 2.17 \[ \frac {2 \left (315 b^{4} B \,x^{5} e^{5}+385 A \,b^{4} e^{5} x^{4}+1540 B a \,b^{3} e^{5} x^{4}-350 B \,b^{4} d \,e^{4} x^{4}+1980 A a \,b^{3} e^{5} x^{3}-440 A \,b^{4} d \,e^{4} x^{3}+2970 B \,a^{2} b^{2} e^{5} x^{3}-1760 B a \,b^{3} d \,e^{4} x^{3}+400 B \,b^{4} d^{2} e^{3} x^{3}+4158 A \,a^{2} b^{2} e^{5} x^{2}-2376 A a \,b^{3} d \,e^{4} x^{2}+528 A \,b^{4} d^{2} e^{3} x^{2}+2772 B \,a^{3} b \,e^{5} x^{2}-3564 B \,a^{2} b^{2} d \,e^{4} x^{2}+2112 B a \,b^{3} d^{2} e^{3} x^{2}-480 B \,b^{4} d^{3} e^{2} x^{2}+4620 A \,a^{3} b \,e^{5} x -5544 A \,a^{2} b^{2} d \,e^{4} x +3168 A a \,b^{3} d^{2} e^{3} x -704 A \,b^{4} d^{3} e^{2} x +1155 B \,a^{4} e^{5} x -3696 B \,a^{3} b d \,e^{4} x +4752 B \,a^{2} b^{2} d^{2} e^{3} x -2816 B a \,b^{3} d^{3} e^{2} x +640 B \,b^{4} d^{4} e x +3465 A \,a^{4} e^{5}-9240 A \,a^{3} b d \,e^{4}+11088 A \,a^{2} b^{2} d^{2} e^{3}-6336 A a \,b^{3} d^{3} e^{2}+1408 A \,b^{4} d^{4} e -2310 B \,a^{4} d \,e^{4}+7392 B \,d^{2} a^{3} b \,e^{3}-9504 B \,d^{3} a^{2} b^{2} e^{2}+5632 B a \,b^{3} d^{4} e -1280 B \,b^{4} d^{5}\right ) \sqrt {e x +d}}{3465 e^{6}} \]

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)^(1/2),x)

[Out]

2/3465*(315*B*b^4*e^5*x^5+385*A*b^4*e^5*x^4+1540*B*a*b^3*e^5*x^4-350*B*b^4*d*e^4*x^4+1980*A*a*b^3*e^5*x^3-440*
A*b^4*d*e^4*x^3+2970*B*a^2*b^2*e^5*x^3-1760*B*a*b^3*d*e^4*x^3+400*B*b^4*d^2*e^3*x^3+4158*A*a^2*b^2*e^5*x^2-237
6*A*a*b^3*d*e^4*x^2+528*A*b^4*d^2*e^3*x^2+2772*B*a^3*b*e^5*x^2-3564*B*a^2*b^2*d*e^4*x^2+2112*B*a*b^3*d^2*e^3*x
^2-480*B*b^4*d^3*e^2*x^2+4620*A*a^3*b*e^5*x-5544*A*a^2*b^2*d*e^4*x+3168*A*a*b^3*d^2*e^3*x-704*A*b^4*d^3*e^2*x+
1155*B*a^4*e^5*x-3696*B*a^3*b*d*e^4*x+4752*B*a^2*b^2*d^2*e^3*x-2816*B*a*b^3*d^3*e^2*x+640*B*b^4*d^4*e*x+3465*A
*a^4*e^5-9240*A*a^3*b*d*e^4+11088*A*a^2*b^2*d^2*e^3-6336*A*a*b^3*d^3*e^2+1408*A*b^4*d^4*e-2310*B*a^4*d*e^4+739
2*B*a^3*b*d^2*e^3-9504*B*a^2*b^2*d^3*e^2+5632*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 = 0.60, size = 409, normalized size = 1.89 \[ \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{4} - 385 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \]

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)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*b^4 - 385*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(9/2) + 990*(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)^(7/2) - 1386*(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)^(5/2) + 1155
*(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)^(3/2) - 3465*(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^6

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mupad [B]  time = 0.06, size = 197, normalized size = 0.91 \[ \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{9\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{3\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{5\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{7\,e^6} \]

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)^(1/2),x)

[Out]

((d + e*x)^(9/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(9*e^6) + (2*(a*e - b*d)^3*(d + e*x)^(3/2)*(4*A*b*e +
B*a*e - 5*B*b*d))/(3*e^6) + (2*B*b^4*(d + e*x)^(11/2))/(11*e^6) + (2*(A*e - B*d)*(a*e - b*d)^4*(d + e*x)^(1/2
))/e^6 + (4*b*(a*e - b*d)^2*(d + e*x)^(5/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d))/(5*e^6) + (4*b^2*(a*e - b*d)*(d + e
*x)^(7/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(7*e^6)

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sympy [A]  time = 127.18, size = 1311, normalized size = 6.07 \[ \text {result too large to display} \]

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)**(1/2),x)

[Out]

Piecewise(((-2*A*a**4*d/sqrt(d + e*x) - 2*A*a**4*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 8*A*a**3*b*d*(-d/sqrt(d
+ e*x) - sqrt(d + e*x))/e - 8*A*a**3*b*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 12*A*
a**2*b**2*d*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 12*A*a**2*b**2*(-d**3/sqrt(d
+ e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 8*A*a*b**3*d*(-d**3/sqrt(d + e
*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 8*A*a*b**3*(d**4/sqrt(d + e*x) +
4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 2*A*b**4*
d*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(
7/2)/7)/e**4 - 2*A*b**4*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d +
e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 2*B*a**4*d*(-d/sqrt(d + e*x) - sqrt(d + e*x
))/e - 2*B*a**4*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 8*B*a**3*b*d*(d**2/sqrt(d +
e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 - 8*B*a**3*b*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) +
d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 - 12*B*a**2*b**2*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x)
+ d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**3 - 12*B*a**2*b**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) -
2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3 - 8*B*a*b**3*d*(d**4/sqrt(d + e*x)
+ 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**4 - 8*B*a*
b**3*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*
(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**4 - 2*B*b**4*d*(-d**5/sqrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d*
*3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7/2)/7 - (d + e*x)**(9/2)/9)/e**5 - 2*B*b**4
*(d**6/sqrt(d + e*x) + 6*d**5*sqrt(d + e*x) - 5*d**4*(d + e*x)**(3/2) + 4*d**3*(d + e*x)**(5/2) - 15*d**2*(d +
e*x)**(7/2)/7 + 2*d*(d + e*x)**(9/2)/3 - (d + e*x)**(11/2)/11)/e**5)/e, Ne(e, 0)), ((A*a**4*x + B*b**4*x**6/6
+ x**5*(A*b**4 + 4*B*a*b**3)/5 + x**4*(4*A*a*b**3 + 6*B*a**2*b**2)/4 + x**3*(6*A*a**2*b**2 + 4*B*a**3*b)/3 +
x**2*(4*A*a**3*b + B*a**4)/2)/sqrt(d), True))

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