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

Optimal. Leaf size=179 \[ -\frac {4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac {30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac {8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac {12 b \sqrt {d+e x} (b d-a e)^5}{e^7}-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7} \]

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Rubi [A]  time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 43} \begin {gather*} -\frac {4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac {30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac {8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac {10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac {12 b \sqrt {d+e x} (b d-a e)^5}{e^7}-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^6)/(e^7*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d + e*x])/e^7 + (10*b^2*(b*d - a*e)^4*(d + e
*x)^(3/2))/e^7 - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2))/e^7 + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^7) -
(4*b^5*(b*d - a*e)*(d + e*x)^(9/2))/(3*e^7) + (2*b^6*(d + e*x)^(11/2))/(11*e^7)

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 \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{3/2}}-\frac {6 b (b d-a e)^5}{e^6 \sqrt {d+e x}}+\frac {15 b^2 (b d-a e)^4 \sqrt {d+e x}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{7/2}}{e^6}+\frac {b^6 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac {2 (b d-a e)^6}{e^7 \sqrt {d+e x}}-\frac {12 b (b d-a e)^5 \sqrt {d+e x}}{e^7}+\frac {10 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac {8 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{7/2}}{7 e^7}-\frac {4 b^5 (b d-a e) (d+e x)^{9/2}}{3 e^7}+\frac {2 b^6 (d+e x)^{11/2}}{11 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 145, normalized size = 0.81 \begin {gather*} \frac {2 \left (-154 b^5 (d+e x)^5 (b d-a e)+495 b^4 (d+e x)^4 (b d-a e)^2-924 b^3 (d+e x)^3 (b d-a e)^3+1155 b^2 (d+e x)^2 (b d-a e)^4-1386 b (d+e x) (b d-a e)^5-231 (b d-a e)^6+21 b^6 (d+e x)^6\right )}{231 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-231*(b*d - a*e)^6 - 1386*b*(b*d - a*e)^5*(d + e*x) + 1155*b^2*(b*d - a*e)^4*(d + e*x)^2 - 924*b^3*(b*d -
a*e)^3*(d + e*x)^3 + 495*b^4*(b*d - a*e)^2*(d + e*x)^4 - 154*b^5*(b*d - a*e)*(d + e*x)^5 + 21*b^6*(d + e*x)^6)
)/(231*e^7*Sqrt[d + e*x])

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IntegrateAlgebraic [B]  time = 0.09, size = 438, normalized size = 2.45 \begin {gather*} \frac {2 \left (-231 a^6 e^6+1386 a^5 b e^5 (d+e x)+1386 a^5 b d e^5-3465 a^4 b^2 d^2 e^4+1155 a^4 b^2 e^4 (d+e x)^2-6930 a^4 b^2 d e^4 (d+e x)+4620 a^3 b^3 d^3 e^3+13860 a^3 b^3 d^2 e^3 (d+e x)+924 a^3 b^3 e^3 (d+e x)^3-4620 a^3 b^3 d e^3 (d+e x)^2-3465 a^2 b^4 d^4 e^2-13860 a^2 b^4 d^3 e^2 (d+e x)+6930 a^2 b^4 d^2 e^2 (d+e x)^2+495 a^2 b^4 e^2 (d+e x)^4-2772 a^2 b^4 d e^2 (d+e x)^3+1386 a b^5 d^5 e+6930 a b^5 d^4 e (d+e x)-4620 a b^5 d^3 e (d+e x)^2+2772 a b^5 d^2 e (d+e x)^3+154 a b^5 e (d+e x)^5-990 a b^5 d e (d+e x)^4-231 b^6 d^6-1386 b^6 d^5 (d+e x)+1155 b^6 d^4 (d+e x)^2-924 b^6 d^3 (d+e x)^3+495 b^6 d^2 (d+e x)^4+21 b^6 (d+e x)^6-154 b^6 d (d+e x)^5\right )}{231 e^7 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-231*b^6*d^6 + 1386*a*b^5*d^5*e - 3465*a^2*b^4*d^4*e^2 + 4620*a^3*b^3*d^3*e^3 - 3465*a^4*b^2*d^2*e^4 + 138
6*a^5*b*d*e^5 - 231*a^6*e^6 - 1386*b^6*d^5*(d + e*x) + 6930*a*b^5*d^4*e*(d + e*x) - 13860*a^2*b^4*d^3*e^2*(d +
e*x) + 13860*a^3*b^3*d^2*e^3*(d + e*x) - 6930*a^4*b^2*d*e^4*(d + e*x) + 1386*a^5*b*e^5*(d + e*x) + 1155*b^6*d
^4*(d + e*x)^2 - 4620*a*b^5*d^3*e*(d + e*x)^2 + 6930*a^2*b^4*d^2*e^2*(d + e*x)^2 - 4620*a^3*b^3*d*e^3*(d + e*x
)^2 + 1155*a^4*b^2*e^4*(d + e*x)^2 - 924*b^6*d^3*(d + e*x)^3 + 2772*a*b^5*d^2*e*(d + e*x)^3 - 2772*a^2*b^4*d*e
^2*(d + e*x)^3 + 924*a^3*b^3*e^3*(d + e*x)^3 + 495*b^6*d^2*(d + e*x)^4 - 990*a*b^5*d*e*(d + e*x)^4 + 495*a^2*b
^4*e^2*(d + e*x)^4 - 154*b^6*d*(d + e*x)^5 + 154*a*b^5*e*(d + e*x)^5 + 21*b^6*(d + e*x)^6))/(231*e^7*Sqrt[d +
e*x])

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fricas [B]  time = 0.41, size = 365, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \, {\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} + {\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \, {\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{231 \, {\left (e^{8} x + d e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^4*e^2 + 14784*a^3*b^3*d^3*e^3 - 9240
*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6 - 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 4
4*a*b^5*d*e^5 + 99*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231*a^3*b^3*e
^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d^3*e^3 + 1584*a^2*b^4*d^2*e^4 - 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)
*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3*b^3*d^2*e^4 + 2310*a^4*b^2*d*e^
5 - 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^8*x + d*e^7)

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giac [B]  time = 0.26, size = 474, normalized size = 2.65 \begin {gather*} \frac {2}{231} \, {\left (21 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{6} e^{70} - 154 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{6} d e^{70} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{6} d^{2} e^{70} - 924 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} d^{3} e^{70} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} d^{4} e^{70} - 1386 \, \sqrt {x e + d} b^{6} d^{5} e^{70} + 154 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{5} e^{71} - 990 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{5} d e^{71} + 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{5} d^{2} e^{71} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{5} d^{3} e^{71} + 6930 \, \sqrt {x e + d} a b^{5} d^{4} e^{71} + 495 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{4} e^{72} - 2772 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{4} d e^{72} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{4} d^{2} e^{72} - 13860 \, \sqrt {x e + d} a^{2} b^{4} d^{3} e^{72} + 924 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{3} e^{73} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{3} d e^{73} + 13860 \, \sqrt {x e + d} a^{3} b^{3} d^{2} e^{73} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{2} e^{74} - 6930 \, \sqrt {x e + d} a^{4} b^{2} d e^{74} + 1386 \, \sqrt {x e + d} a^{5} b e^{75}\right )} e^{\left (-77\right )} - \frac {2 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{\sqrt {x e + d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/231*(21*(x*e + d)^(11/2)*b^6*e^70 - 154*(x*e + d)^(9/2)*b^6*d*e^70 + 495*(x*e + d)^(7/2)*b^6*d^2*e^70 - 924*
(x*e + d)^(5/2)*b^6*d^3*e^70 + 1155*(x*e + d)^(3/2)*b^6*d^4*e^70 - 1386*sqrt(x*e + d)*b^6*d^5*e^70 + 154*(x*e
+ d)^(9/2)*a*b^5*e^71 - 990*(x*e + d)^(7/2)*a*b^5*d*e^71 + 2772*(x*e + d)^(5/2)*a*b^5*d^2*e^71 - 4620*(x*e + d
)^(3/2)*a*b^5*d^3*e^71 + 6930*sqrt(x*e + d)*a*b^5*d^4*e^71 + 495*(x*e + d)^(7/2)*a^2*b^4*e^72 - 2772*(x*e + d)
^(5/2)*a^2*b^4*d*e^72 + 6930*(x*e + d)^(3/2)*a^2*b^4*d^2*e^72 - 13860*sqrt(x*e + d)*a^2*b^4*d^3*e^72 + 924*(x*
e + d)^(5/2)*a^3*b^3*e^73 - 4620*(x*e + d)^(3/2)*a^3*b^3*d*e^73 + 13860*sqrt(x*e + d)*a^3*b^3*d^2*e^73 + 1155*
(x*e + d)^(3/2)*a^4*b^2*e^74 - 6930*sqrt(x*e + d)*a^4*b^2*d*e^74 + 1386*sqrt(x*e + d)*a^5*b*e^75)*e^(-77) - 2*
(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*
e^6)*e^(-7)/sqrt(x*e + d)

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maple [B]  time = 0.05, size = 377, normalized size = 2.11 \begin {gather*} -\frac {2 \left (-21 b^{6} e^{6} x^{6}-154 a \,b^{5} e^{6} x^{5}+28 b^{6} d \,e^{5} x^{5}-495 a^{2} b^{4} e^{6} x^{4}+220 a \,b^{5} d \,e^{5} x^{4}-40 b^{6} d^{2} e^{4} x^{4}-924 a^{3} b^{3} e^{6} x^{3}+792 a^{2} b^{4} d \,e^{5} x^{3}-352 a \,b^{5} d^{2} e^{4} x^{3}+64 b^{6} d^{3} e^{3} x^{3}-1155 a^{4} b^{2} e^{6} x^{2}+1848 a^{3} b^{3} d \,e^{5} x^{2}-1584 a^{2} b^{4} d^{2} e^{4} x^{2}+704 a \,b^{5} d^{3} e^{3} x^{2}-128 b^{6} d^{4} e^{2} x^{2}-1386 a^{5} b \,e^{6} x +4620 a^{4} b^{2} d \,e^{5} x -7392 a^{3} b^{3} d^{2} e^{4} x +6336 a^{2} b^{4} d^{3} e^{3} x -2816 a \,b^{5} d^{4} e^{2} x +512 b^{6} d^{5} e x +231 a^{6} e^{6}-2772 a^{5} b d \,e^{5}+9240 a^{4} b^{2} d^{2} e^{4}-14784 a^{3} b^{3} d^{3} e^{3}+12672 a^{2} b^{4} d^{4} e^{2}-5632 a \,b^{5} d^{5} e +1024 b^{6} d^{6}\right )}{231 \sqrt {e x +d}\, e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x)

[Out]

-2/231*(-21*b^6*e^6*x^6-154*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-495*a^2*b^4*e^6*x^4+220*a*b^5*d*e^5*x^4-40*b^6*d^2*
e^4*x^4-924*a^3*b^3*e^6*x^3+792*a^2*b^4*d*e^5*x^3-352*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-1155*a^4*b^2*e^6*x^
2+1848*a^3*b^3*d*e^5*x^2-1584*a^2*b^4*d^2*e^4*x^2+704*a*b^5*d^3*e^3*x^2-128*b^6*d^4*e^2*x^2-1386*a^5*b*e^6*x+4
620*a^4*b^2*d*e^5*x-7392*a^3*b^3*d^2*e^4*x+6336*a^2*b^4*d^3*e^3*x-2816*a*b^5*d^4*e^2*x+512*b^6*d^5*e*x+231*a^6
*e^6-2772*a^5*b*d*e^5+9240*a^4*b^2*d^2*e^4-14784*a^3*b^3*d^3*e^3+12672*a^2*b^4*d^4*e^2-5632*a*b^5*d^5*e+1024*b
^6*d^6)/(e*x+d)^(1/2)/e^7

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maxima [B]  time = 1.03, size = 358, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (\frac {21 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{6} - 154 \, {\left (b^{6} d - a b^{5} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 495 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 924 \, {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 1386 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \sqrt {e x + d}}{e^{6}} - \frac {231 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}}{\sqrt {e x + d} e^{6}}\right )}}{231 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/231*((21*(e*x + d)^(11/2)*b^6 - 154*(b^6*d - a*b^5*e)*(e*x + d)^(9/2) + 495*(b^6*d^2 - 2*a*b^5*d*e + a^2*b^4
*e^2)*(e*x + d)^(7/2) - 924*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d)^(5/2) + 1155*(
b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(3/2) - 1386*(b^6*d^5 -
5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*sqrt(e*x + d))/e^6 - 2
31*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a
^6*e^6)/(sqrt(e*x + d)*e^6))/e

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mupad [B]  time = 0.55, size = 231, normalized size = 1.29 \begin {gather*} \frac {2\,b^6\,{\left (d+e\,x\right )}^{11/2}}{11\,e^7}-\frac {2\,a^6\,e^6-12\,a^5\,b\,d\,e^5+30\,a^4\,b^2\,d^2\,e^4-40\,a^3\,b^3\,d^3\,e^3+30\,a^2\,b^4\,d^4\,e^2-12\,a\,b^5\,d^5\,e+2\,b^6\,d^6}{e^7\,\sqrt {d+e\,x}}-\frac {\left (12\,b^6\,d-12\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}+\frac {10\,b^2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{e^7}+\frac {8\,b^3\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^7}+\frac {30\,b^4\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}+\frac {12\,b\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^(3/2),x)

[Out]

(2*b^6*(d + e*x)^(11/2))/(11*e^7) - (2*a^6*e^6 + 2*b^6*d^6 + 30*a^2*b^4*d^4*e^2 - 40*a^3*b^3*d^3*e^3 + 30*a^4*
b^2*d^2*e^4 - 12*a*b^5*d^5*e - 12*a^5*b*d*e^5)/(e^7*(d + e*x)^(1/2)) - ((12*b^6*d - 12*a*b^5*e)*(d + e*x)^(9/2
))/(9*e^7) + (10*b^2*(a*e - b*d)^4*(d + e*x)^(3/2))/e^7 + (8*b^3*(a*e - b*d)^3*(d + e*x)^(5/2))/e^7 + (30*b^4*
(a*e - b*d)^2*(d + e*x)^(7/2))/(7*e^7) + (12*b*(a*e - b*d)^5*(d + e*x)^(1/2))/e^7

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sympy [A]  time = 69.70, size = 333, normalized size = 1.86 \begin {gather*} \frac {2 b^{6} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (30 a^{4} b^{2} e^{4} - 120 a^{3} b^{3} d e^{3} + 180 a^{2} b^{4} d^{2} e^{2} - 120 a b^{5} d^{3} e + 30 b^{6} d^{4}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (12 a^{5} b e^{5} - 60 a^{4} b^{2} d e^{4} + 120 a^{3} b^{3} d^{2} e^{3} - 120 a^{2} b^{4} d^{3} e^{2} + 60 a b^{5} d^{4} e - 12 b^{6} d^{5}\right )}{e^{7}} - \frac {2 \left (a e - b d\right )^{6}}{e^{7} \sqrt {d + e x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**(3/2),x)

[Out]

2*b**6*(d + e*x)**(11/2)/(11*e**7) + (d + e*x)**(9/2)*(12*a*b**5*e - 12*b**6*d)/(9*e**7) + (d + e*x)**(7/2)*(3
0*a**2*b**4*e**2 - 60*a*b**5*d*e + 30*b**6*d**2)/(7*e**7) + (d + e*x)**(5/2)*(40*a**3*b**3*e**3 - 120*a**2*b**
4*d*e**2 + 120*a*b**5*d**2*e - 40*b**6*d**3)/(5*e**7) + (d + e*x)**(3/2)*(30*a**4*b**2*e**4 - 120*a**3*b**3*d*
e**3 + 180*a**2*b**4*d**2*e**2 - 120*a*b**5*d**3*e + 30*b**6*d**4)/(3*e**7) + sqrt(d + e*x)*(12*a**5*b*e**5 -
60*a**4*b**2*d*e**4 + 120*a**3*b**3*d**2*e**3 - 120*a**2*b**4*d**3*e**2 + 60*a*b**5*d**4*e - 12*b**6*d**5)/e**
7 - 2*(a*e - b*d)**6/(e**7*sqrt(d + e*x))

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