3.19.28 \(\int (a+b x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=156 \[ -\frac {10 b^4 (d+e x)^{11/2} (b d-a e)}{11 e^6}+\frac {20 b^3 (d+e x)^{9/2} (b d-a e)^2}{9 e^6}-\frac {20 b^2 (d+e x)^{7/2} (b d-a e)^3}{7 e^6}+\frac {2 b (d+e x)^{5/2} (b d-a e)^4}{e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^5}{3 e^6}+\frac {2 b^5 (d+e x)^{13/2}}{13 e^6} \]

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Rubi [A]  time = 0.05, antiderivative size = 156, 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, 43} \begin {gather*} -\frac {10 b^4 (d+e x)^{11/2} (b d-a e)}{11 e^6}+\frac {20 b^3 (d+e x)^{9/2} (b d-a e)^2}{9 e^6}-\frac {20 b^2 (d+e x)^{7/2} (b d-a e)^3}{7 e^6}+\frac {2 b (d+e x)^{5/2} (b d-a e)^4}{e^6}-\frac {2 (d+e x)^{3/2} (b d-a e)^5}{3 e^6}+\frac {2 b^5 (d+e x)^{13/2}}{13 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(3/2))/(3*e^6) + (2*b*(b*d - a*e)^4*(d + e*x)^(5/2))/e^6 - (20*b^2*(b*d - a*e)^3*(
d + e*x)^(7/2))/(7*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(11/2
))/(11*e^6) + (2*b^5*(d + e*x)^(13/2))/(13*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^5 \sqrt {d+e x}}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{3/2}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{5/2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{7/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{9/2}}{e^5}+\frac {b^5 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^5 (d+e x)^{3/2}}{3 e^6}+\frac {2 b (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{7/2}}{7 e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{9/2}}{9 e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{11/2}}{11 e^6}+\frac {2 b^5 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 123, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-4095 b^4 (d+e x)^4 (b d-a e)+10010 b^3 (d+e x)^3 (b d-a e)^2-12870 b^2 (d+e x)^2 (b d-a e)^3+9009 b (d+e x) (b d-a e)^4-3003 (b d-a e)^5+693 b^5 (d+e x)^5\right )}{9009 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-3003*(b*d - a*e)^5 + 9009*b*(b*d - a*e)^4*(d + e*x) - 12870*b^2*(b*d - a*e)^3*(d + e*x)^2
+ 10010*b^3*(b*d - a*e)^2*(d + e*x)^3 - 4095*b^4*(b*d - a*e)*(d + e*x)^4 + 693*b^5*(d + e*x)^5))/(9009*e^6)

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IntegrateAlgebraic [B]  time = 0.10, size = 315, normalized size = 2.02 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (3003 a^5 e^5+9009 a^4 b e^4 (d+e x)-15015 a^4 b d e^4+30030 a^3 b^2 d^2 e^3+12870 a^3 b^2 e^3 (d+e x)^2-36036 a^3 b^2 d e^3 (d+e x)-30030 a^2 b^3 d^3 e^2+54054 a^2 b^3 d^2 e^2 (d+e x)+10010 a^2 b^3 e^2 (d+e x)^3-38610 a^2 b^3 d e^2 (d+e x)^2+15015 a b^4 d^4 e-36036 a b^4 d^3 e (d+e x)+38610 a b^4 d^2 e (d+e x)^2+4095 a b^4 e (d+e x)^4-20020 a b^4 d e (d+e x)^3-3003 b^5 d^5+9009 b^5 d^4 (d+e x)-12870 b^5 d^3 (d+e x)^2+10010 b^5 d^2 (d+e x)^3+693 b^5 (d+e x)^5-4095 b^5 d (d+e x)^4\right )}{9009 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(-3003*b^5*d^5 + 15015*a*b^4*d^4*e - 30030*a^2*b^3*d^3*e^2 + 30030*a^3*b^2*d^2*e^3 - 15015*
a^4*b*d*e^4 + 3003*a^5*e^5 + 9009*b^5*d^4*(d + e*x) - 36036*a*b^4*d^3*e*(d + e*x) + 54054*a^2*b^3*d^2*e^2*(d +
e*x) - 36036*a^3*b^2*d*e^3*(d + e*x) + 9009*a^4*b*e^4*(d + e*x) - 12870*b^5*d^3*(d + e*x)^2 + 38610*a*b^4*d^2
*e*(d + e*x)^2 - 38610*a^2*b^3*d*e^2*(d + e*x)^2 + 12870*a^3*b^2*e^3*(d + e*x)^2 + 10010*b^5*d^2*(d + e*x)^3 -
20020*a*b^4*d*e*(d + e*x)^3 + 10010*a^2*b^3*e^2*(d + e*x)^3 - 4095*b^5*d*(d + e*x)^4 + 4095*a*b^4*e*(d + e*x)
^4 + 693*b^5*(d + e*x)^5))/(9009*e^6)

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fricas [B]  time = 0.43, size = 338, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{6}} \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)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*
a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*
a^2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^
5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e
- 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(
e*x + d)/e^6

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giac [B]  time = 0.19, size = 678, normalized size = 4.35 \begin {gather*} \frac {2}{9009} \, {\left (15015 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{4} b d e^{\left (-1\right )} + 6006 \, {\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^{3} b^{2} d e^{\left (-2\right )} + 2574 \, {\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^{2} b^{3} d e^{\left (-3\right )} + 143 \, {\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} d e^{\left (-4\right )} + 13 \, {\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^{5} d e^{\left (-5\right )} + 3003 \, {\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^{4} b e^{\left (-1\right )} + 2574 \, {\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^{3} b^{2} e^{\left (-2\right )} + 286 \, {\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^{2} b^{3} e^{\left (-3\right )} + 65 \, {\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 )} a b^{4} e^{\left (-4\right )} + 3 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} b^{5} e^{\left (-5\right )} + 9009 \, \sqrt {x e + d} a^{5} d + 3003 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{5}\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)^(1/2),x, algorithm="giac")

[Out]

2/9009*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^4*b*d*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3
/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b^2*d*e^(-2) + 2574*(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^2*b^3*d*e^(-3) + 143*(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*d*e^(-4) + 13*(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^5*d*e^(-5) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^4*b
*e^(-1) + 2574*(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^3*
b^2*e^(-2) + 286*(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^2*b^3*e^(-3) + 65*(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)*a*b^4*e^(-4) + 3*(231*
(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e +
d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^5*e^(-5) + 9009*sqrt(x*e + d)*a^5*d + 300
3*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^5)*e^(-1)

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maple [B]  time = 0.05, size = 273, normalized size = 1.75 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (693 b^{5} e^{5} x^{5}+4095 a \,b^{4} e^{5} x^{4}-630 b^{5} d \,e^{4} x^{4}+10010 a^{2} b^{3} e^{5} x^{3}-3640 a \,b^{4} d \,e^{4} x^{3}+560 b^{5} d^{2} e^{3} x^{3}+12870 a^{3} b^{2} e^{5} x^{2}-8580 a^{2} b^{3} d \,e^{4} x^{2}+3120 a \,b^{4} d^{2} e^{3} x^{2}-480 b^{5} d^{3} e^{2} x^{2}+9009 a^{4} b \,e^{5} x -10296 a^{3} b^{2} d \,e^{4} x +6864 a^{2} b^{3} d^{2} e^{3} x -2496 a \,b^{4} d^{3} e^{2} x +384 b^{5} d^{4} e x +3003 a^{5} e^{5}-6006 a^{4} b d \,e^{4}+6864 a^{3} b^{2} d^{2} e^{3}-4576 a^{2} b^{3} d^{3} e^{2}+1664 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{9009 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)^(1/2),x)

[Out]

2/9009*(e*x+d)^(3/2)*(693*b^5*e^5*x^5+4095*a*b^4*e^5*x^4-630*b^5*d*e^4*x^4+10010*a^2*b^3*e^5*x^3-3640*a*b^4*d*
e^4*x^3+560*b^5*d^2*e^3*x^3+12870*a^3*b^2*e^5*x^2-8580*a^2*b^3*d*e^4*x^2+3120*a*b^4*d^2*e^3*x^2-480*b^5*d^3*e^
2*x^2+9009*a^4*b*e^5*x-10296*a^3*b^2*d*e^4*x+6864*a^2*b^3*d^2*e^3*x-2496*a*b^4*d^3*e^2*x+384*b^5*d^4*e*x+3003*
a^5*e^5-6006*a^4*b*d*e^4+6864*a^3*b^2*d^2*e^3-4576*a^2*b^3*d^3*e^2+1664*a*b^4*d^4*e-256*b^5*d^5)/e^6

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maxima [A]  time = 0.49, size = 259, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{5} - 4095 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 12870 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 3003 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{9009 \, e^{6}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*(e*x + d)^(13/2)*b^5 - 4095*(b^5*d - a*b^4*e)*(e*x + d)^(11/2) + 10010*(b^5*d^2 - 2*a*b^4*d*e + a^
2*b^3*e^2)*(e*x + d)^(9/2) - 12870*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(7/2) +
9009*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^3 + a^4*b*e^4)*(e*x + d)^(5/2) - 3003*(b^5*
d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(3/2))/e^6

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mupad [B]  time = 0.05, size = 137, normalized size = 0.88 \begin {gather*} \frac {2\,b^5\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{5/2}}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(d + e*x)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

(2*b^5*(d + e*x)^(13/2))/(13*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(11/2))/(11*e^6) + (2*(a*e - b*d)^5*(d
+ e*x)^(3/2))/(3*e^6) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(7/2))/(7*e^6) + (20*b^3*(a*e - b*d)^2*(d + e*x)^(9/2)
)/(9*e^6) + (2*b*(a*e - b*d)^4*(d + e*x)^(5/2))/e^6

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sympy [B]  time = 7.70, size = 314, normalized size = 2.01 \begin {gather*} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (5 a b^{4} e - 5 b^{5} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{3 e^{5}}\right )}{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)**(1/2),x)

[Out]

2*(b**5*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*a*b**4*e - 5*b**5*d)/(11*e**5) + (d + e*x)**(9/2)*(
10*a**2*b**3*e**2 - 20*a*b**4*d*e + 10*b**5*d**2)/(9*e**5) + (d + e*x)**(7/2)*(10*a**3*b**2*e**3 - 30*a**2*b**
3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(7*e**5) + (d + e*x)**(5/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 +
30*a**2*b**3*d**2*e**2 - 20*a*b**4*d**3*e + 5*b**5*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**5*e**5 - 5*a**4*b*d*
e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/(3*e**5))/e

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