3.19.39 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ -\frac {14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac {6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac {14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac {70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 \sqrt {d+e x} (b d-a e)^5}{e^8}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8} \]

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Rubi [A]  time = 0.08, antiderivative size = 208, 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 {14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac {6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac {14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac {70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 \sqrt {d+e x} (b d-a e)^5}{e^8}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}+\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(b*d - a*e)^7)/(3*e^8*(d + e*x)^(3/2)) - (14*b*(b*d - a*e)^6)/(e^8*Sqrt[d + e*x]) - (42*b^2*(b*d - a*e)^5*S
qrt[d + e*x])/e^8 + (70*b^3*(b*d - a*e)^4*(d + e*x)^(3/2))/(3*e^8) - (14*b^4*(b*d - a*e)^3*(d + e*x)^(5/2))/e^
8 + (6*b^5*(b*d - a*e)^2*(d + e*x)^(7/2))/e^8 - (14*b^6*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^8) + (2*b^7*(d + e*x
)^(11/2))/(11*e^8)

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 {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \frac {(a+b x)^7}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac {(-b d+a e)^7}{e^7 (d+e x)^{5/2}}+\frac {7 b (b d-a e)^6}{e^7 (d+e x)^{3/2}}-\frac {21 b^2 (b d-a e)^5}{e^7 \sqrt {d+e x}}+\frac {35 b^3 (b d-a e)^4 \sqrt {d+e x}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{3/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{7/2}}{e^7}+\frac {b^7 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac {2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}-\frac {14 b (b d-a e)^6}{e^8 \sqrt {d+e x}}-\frac {42 b^2 (b d-a e)^5 \sqrt {d+e x}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{3/2}}{3 e^8}-\frac {14 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^8}+\frac {6 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{9/2}}{9 e^8}+\frac {2 b^7 (d+e x)^{11/2}}{11 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 167, normalized size = 0.80 \begin {gather*} \frac {2 \left (-77 b^6 (d+e x)^6 (b d-a e)+297 b^5 (d+e x)^5 (b d-a e)^2-693 b^4 (d+e x)^4 (b d-a e)^3+1155 b^3 (d+e x)^3 (b d-a e)^4-2079 b^2 (d+e x)^2 (b d-a e)^5-693 b (d+e x) (b d-a e)^6+33 (b d-a e)^7+9 b^7 (d+e x)^7\right )}{99 e^8 (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)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(33*(b*d - a*e)^7 - 693*b*(b*d - a*e)^6*(d + e*x) - 2079*b^2*(b*d - a*e)^5*(d + e*x)^2 + 1155*b^3*(b*d - a*
e)^4*(d + e*x)^3 - 693*b^4*(b*d - a*e)^3*(d + e*x)^4 + 297*b^5*(b*d - a*e)^2*(d + e*x)^5 - 77*b^6*(b*d - a*e)*
(d + e*x)^6 + 9*b^7*(d + e*x)^7))/(99*e^8*(d + e*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.12, size = 582, normalized size = 2.80 \begin {gather*} \frac {2 \left (-33 a^7 e^7-693 a^6 b e^6 (d+e x)+231 a^6 b d e^6-693 a^5 b^2 d^2 e^5+2079 a^5 b^2 e^5 (d+e x)^2+4158 a^5 b^2 d e^5 (d+e x)+1155 a^4 b^3 d^3 e^4-10395 a^4 b^3 d^2 e^4 (d+e x)+1155 a^4 b^3 e^4 (d+e x)^3-10395 a^4 b^3 d e^4 (d+e x)^2-1155 a^3 b^4 d^4 e^3+13860 a^3 b^4 d^3 e^3 (d+e x)+20790 a^3 b^4 d^2 e^3 (d+e x)^2+693 a^3 b^4 e^3 (d+e x)^4-4620 a^3 b^4 d e^3 (d+e x)^3+693 a^2 b^5 d^5 e^2-10395 a^2 b^5 d^4 e^2 (d+e x)-20790 a^2 b^5 d^3 e^2 (d+e x)^2+6930 a^2 b^5 d^2 e^2 (d+e x)^3+297 a^2 b^5 e^2 (d+e x)^5-2079 a^2 b^5 d e^2 (d+e x)^4-231 a b^6 d^6 e+4158 a b^6 d^5 e (d+e x)+10395 a b^6 d^4 e (d+e x)^2-4620 a b^6 d^3 e (d+e x)^3+2079 a b^6 d^2 e (d+e x)^4+77 a b^6 e (d+e x)^6-594 a b^6 d e (d+e x)^5+33 b^7 d^7-693 b^7 d^6 (d+e x)-2079 b^7 d^5 (d+e x)^2+1155 b^7 d^4 (d+e x)^3-693 b^7 d^3 (d+e x)^4+297 b^7 d^2 (d+e x)^5+9 b^7 (d+e x)^7-77 b^7 d (d+e x)^6\right )}{99 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(33*b^7*d^7 - 231*a*b^6*d^6*e + 693*a^2*b^5*d^5*e^2 - 1155*a^3*b^4*d^4*e^3 + 1155*a^4*b^3*d^3*e^4 - 693*a^5
*b^2*d^2*e^5 + 231*a^6*b*d*e^6 - 33*a^7*e^7 - 693*b^7*d^6*(d + e*x) + 4158*a*b^6*d^5*e*(d + e*x) - 10395*a^2*b
^5*d^4*e^2*(d + e*x) + 13860*a^3*b^4*d^3*e^3*(d + e*x) - 10395*a^4*b^3*d^2*e^4*(d + e*x) + 4158*a^5*b^2*d*e^5*
(d + e*x) - 693*a^6*b*e^6*(d + e*x) - 2079*b^7*d^5*(d + e*x)^2 + 10395*a*b^6*d^4*e*(d + e*x)^2 - 20790*a^2*b^5
*d^3*e^2*(d + e*x)^2 + 20790*a^3*b^4*d^2*e^3*(d + e*x)^2 - 10395*a^4*b^3*d*e^4*(d + e*x)^2 + 2079*a^5*b^2*e^5*
(d + e*x)^2 + 1155*b^7*d^4*(d + e*x)^3 - 4620*a*b^6*d^3*e*(d + e*x)^3 + 6930*a^2*b^5*d^2*e^2*(d + e*x)^3 - 462
0*a^3*b^4*d*e^3*(d + e*x)^3 + 1155*a^4*b^3*e^4*(d + e*x)^3 - 693*b^7*d^3*(d + e*x)^4 + 2079*a*b^6*d^2*e*(d + e
*x)^4 - 2079*a^2*b^5*d*e^2*(d + e*x)^4 + 693*a^3*b^4*e^3*(d + e*x)^4 + 297*b^7*d^2*(d + e*x)^5 - 594*a*b^6*d*e
*(d + e*x)^5 + 297*a^2*b^5*e^2*(d + e*x)^5 - 77*b^7*d*(d + e*x)^6 + 77*a*b^6*e*(d + e*x)^6 + 9*b^7*(d + e*x)^7
))/(99*e^8*(d + e*x)^(3/2))

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fricas [B]  time = 0.43, size = 484, normalized size = 2.33 \begin {gather*} \frac {2 \, {\left (9 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 11264 \, a b^{6} d^{6} e - 25344 \, a^{2} b^{5} d^{5} e^{2} + 29568 \, a^{3} b^{4} d^{4} e^{3} - 18480 \, a^{4} b^{3} d^{3} e^{4} + 5544 \, a^{5} b^{2} d^{2} e^{5} - 462 \, a^{6} b d e^{6} - 33 \, a^{7} e^{7} - 7 \, {\left (2 \, b^{7} d e^{6} - 11 \, a b^{6} e^{7}\right )} x^{6} + 3 \, {\left (8 \, b^{7} d^{2} e^{5} - 44 \, a b^{6} d e^{6} + 99 \, a^{2} b^{5} e^{7}\right )} x^{5} - 3 \, {\left (16 \, b^{7} d^{3} e^{4} - 88 \, a b^{6} d^{2} e^{5} + 198 \, a^{2} b^{5} d e^{6} - 231 \, a^{3} b^{4} e^{7}\right )} x^{4} + {\left (128 \, b^{7} d^{4} e^{3} - 704 \, a b^{6} d^{3} e^{4} + 1584 \, a^{2} b^{5} d^{2} e^{5} - 1848 \, a^{3} b^{4} d e^{6} + 1155 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1408 \, a b^{6} d^{4} e^{3} + 3168 \, a^{2} b^{5} d^{3} e^{4} - 3696 \, a^{3} b^{4} d^{2} e^{5} + 2310 \, a^{4} b^{3} d e^{6} - 693 \, a^{5} b^{2} e^{7}\right )} x^{2} - 3 \, {\left (1024 \, b^{7} d^{6} e - 5632 \, a b^{6} d^{5} e^{2} + 12672 \, a^{2} b^{5} d^{4} e^{3} - 14784 \, a^{3} b^{4} d^{3} e^{4} + 9240 \, a^{4} b^{3} d^{2} e^{5} - 2772 \, a^{5} b^{2} d e^{6} + 231 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{99 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\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)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/99*(9*b^7*e^7*x^7 - 2048*b^7*d^7 + 11264*a*b^6*d^6*e - 25344*a^2*b^5*d^5*e^2 + 29568*a^3*b^4*d^4*e^3 - 18480
*a^4*b^3*d^3*e^4 + 5544*a^5*b^2*d^2*e^5 - 462*a^6*b*d*e^6 - 33*a^7*e^7 - 7*(2*b^7*d*e^6 - 11*a*b^6*e^7)*x^6 +
3*(8*b^7*d^2*e^5 - 44*a*b^6*d*e^6 + 99*a^2*b^5*e^7)*x^5 - 3*(16*b^7*d^3*e^4 - 88*a*b^6*d^2*e^5 + 198*a^2*b^5*d
*e^6 - 231*a^3*b^4*e^7)*x^4 + (128*b^7*d^4*e^3 - 704*a*b^6*d^3*e^4 + 1584*a^2*b^5*d^2*e^5 - 1848*a^3*b^4*d*e^6
+ 1155*a^4*b^3*e^7)*x^3 - 3*(256*b^7*d^5*e^2 - 1408*a*b^6*d^4*e^3 + 3168*a^2*b^5*d^3*e^4 - 3696*a^3*b^4*d^2*e
^5 + 2310*a^4*b^3*d*e^6 - 693*a^5*b^2*e^7)*x^2 - 3*(1024*b^7*d^6*e - 5632*a*b^6*d^5*e^2 + 12672*a^2*b^5*d^4*e^
3 - 14784*a^3*b^4*d^3*e^4 + 9240*a^4*b^3*d^2*e^5 - 2772*a^5*b^2*d*e^6 + 231*a^6*b*e^7)*x)*sqrt(e*x + d)/(e^10*
x^2 + 2*d*e^9*x + d^2*e^8)

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giac [B]  time = 0.34, size = 609, normalized size = 2.93 \begin {gather*} \frac {2}{99} \, {\left (9 \, {\left (x e + d\right )}^{\frac {11}{2}} b^{7} e^{80} - 77 \, {\left (x e + d\right )}^{\frac {9}{2}} b^{7} d e^{80} + 297 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{7} d^{2} e^{80} - 693 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{7} d^{3} e^{80} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} d^{4} e^{80} - 2079 \, \sqrt {x e + d} b^{7} d^{5} e^{80} + 77 \, {\left (x e + d\right )}^{\frac {9}{2}} a b^{6} e^{81} - 594 \, {\left (x e + d\right )}^{\frac {7}{2}} a b^{6} d e^{81} + 2079 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{6} d^{2} e^{81} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{6} d^{3} e^{81} + 10395 \, \sqrt {x e + d} a b^{6} d^{4} e^{81} + 297 \, {\left (x e + d\right )}^{\frac {7}{2}} a^{2} b^{5} e^{82} - 2079 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{5} d e^{82} + 6930 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{2} e^{82} - 20790 \, \sqrt {x e + d} a^{2} b^{5} d^{3} e^{82} + 693 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{4} e^{83} - 4620 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{4} d e^{83} + 20790 \, \sqrt {x e + d} a^{3} b^{4} d^{2} e^{83} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b^{3} e^{84} - 10395 \, \sqrt {x e + d} a^{4} b^{3} d e^{84} + 2079 \, \sqrt {x e + d} a^{5} b^{2} e^{85}\right )} e^{\left (-88\right )} - \frac {2 \, {\left (21 \, {\left (x e + d\right )} b^{7} d^{6} - b^{7} d^{7} - 126 \, {\left (x e + d\right )} a b^{6} d^{5} e + 7 \, a b^{6} d^{6} e + 315 \, {\left (x e + d\right )} a^{2} b^{5} d^{4} e^{2} - 21 \, a^{2} b^{5} d^{5} e^{2} - 420 \, {\left (x e + d\right )} a^{3} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{4} d^{4} e^{3} + 315 \, {\left (x e + d\right )} a^{4} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{4} - 126 \, {\left (x e + d\right )} a^{5} b^{2} d e^{5} + 21 \, a^{5} b^{2} d^{2} e^{5} + 21 \, {\left (x e + d\right )} a^{6} b e^{6} - 7 \, a^{6} b d e^{6} + a^{7} e^{7}\right )} e^{\left (-8\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)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/99*(9*(x*e + d)^(11/2)*b^7*e^80 - 77*(x*e + d)^(9/2)*b^7*d*e^80 + 297*(x*e + d)^(7/2)*b^7*d^2*e^80 - 693*(x*
e + d)^(5/2)*b^7*d^3*e^80 + 1155*(x*e + d)^(3/2)*b^7*d^4*e^80 - 2079*sqrt(x*e + d)*b^7*d^5*e^80 + 77*(x*e + d)
^(9/2)*a*b^6*e^81 - 594*(x*e + d)^(7/2)*a*b^6*d*e^81 + 2079*(x*e + d)^(5/2)*a*b^6*d^2*e^81 - 4620*(x*e + d)^(3
/2)*a*b^6*d^3*e^81 + 10395*sqrt(x*e + d)*a*b^6*d^4*e^81 + 297*(x*e + d)^(7/2)*a^2*b^5*e^82 - 2079*(x*e + d)^(5
/2)*a^2*b^5*d*e^82 + 6930*(x*e + d)^(3/2)*a^2*b^5*d^2*e^82 - 20790*sqrt(x*e + d)*a^2*b^5*d^3*e^82 + 693*(x*e +
d)^(5/2)*a^3*b^4*e^83 - 4620*(x*e + d)^(3/2)*a^3*b^4*d*e^83 + 20790*sqrt(x*e + d)*a^3*b^4*d^2*e^83 + 1155*(x*
e + d)^(3/2)*a^4*b^3*e^84 - 10395*sqrt(x*e + d)*a^4*b^3*d*e^84 + 2079*sqrt(x*e + d)*a^5*b^2*e^85)*e^(-88) - 2/
3*(21*(x*e + d)*b^7*d^6 - b^7*d^7 - 126*(x*e + d)*a*b^6*d^5*e + 7*a*b^6*d^6*e + 315*(x*e + d)*a^2*b^5*d^4*e^2
- 21*a^2*b^5*d^5*e^2 - 420*(x*e + d)*a^3*b^4*d^3*e^3 + 35*a^3*b^4*d^4*e^3 + 315*(x*e + d)*a^4*b^3*d^2*e^4 - 35
*a^4*b^3*d^3*e^4 - 126*(x*e + d)*a^5*b^2*d*e^5 + 21*a^5*b^2*d^2*e^5 + 21*(x*e + d)*a^6*b*e^6 - 7*a^6*b*d*e^6 +
a^7*e^7)*e^(-8)/(x*e + d)^(3/2)

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maple [B]  time = 0.05, size = 498, normalized size = 2.39 \begin {gather*} -\frac {2 \left (-9 b^{7} x^{7} e^{7}-77 a \,b^{6} e^{7} x^{6}+14 b^{7} d \,e^{6} x^{6}-297 a^{2} b^{5} e^{7} x^{5}+132 a \,b^{6} d \,e^{6} x^{5}-24 b^{7} d^{2} e^{5} x^{5}-693 a^{3} b^{4} e^{7} x^{4}+594 a^{2} b^{5} d \,e^{6} x^{4}-264 a \,b^{6} d^{2} e^{5} x^{4}+48 b^{7} d^{3} e^{4} x^{4}-1155 a^{4} b^{3} e^{7} x^{3}+1848 a^{3} b^{4} d \,e^{6} x^{3}-1584 a^{2} b^{5} d^{2} e^{5} x^{3}+704 a \,b^{6} d^{3} e^{4} x^{3}-128 b^{7} d^{4} e^{3} x^{3}-2079 a^{5} b^{2} e^{7} x^{2}+6930 a^{4} b^{3} d \,e^{6} x^{2}-11088 a^{3} b^{4} d^{2} e^{5} x^{2}+9504 a^{2} b^{5} d^{3} e^{4} x^{2}-4224 a \,b^{6} d^{4} e^{3} x^{2}+768 b^{7} d^{5} e^{2} x^{2}+693 a^{6} b \,e^{7} x -8316 a^{5} b^{2} d \,e^{6} x +27720 a^{4} b^{3} d^{2} e^{5} x -44352 a^{3} b^{4} d^{3} e^{4} x +38016 a^{2} b^{5} d^{4} e^{3} x -16896 a \,b^{6} d^{5} e^{2} x +3072 b^{7} d^{6} e x +33 a^{7} e^{7}+462 a^{6} b d \,e^{6}-5544 a^{5} b^{2} d^{2} e^{5}+18480 a^{4} b^{3} d^{3} e^{4}-29568 a^{3} b^{4} d^{4} e^{3}+25344 a^{2} b^{5} d^{5} e^{2}-11264 a \,b^{6} d^{6} e +2048 b^{7} d^{7}\right )}{99 \left (e x +d \right )^{\frac {3}{2}} e^{8}} \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)^3/(e*x+d)^(5/2),x)

[Out]

-2/99*(-9*b^7*e^7*x^7-77*a*b^6*e^7*x^6+14*b^7*d*e^6*x^6-297*a^2*b^5*e^7*x^5+132*a*b^6*d*e^6*x^5-24*b^7*d^2*e^5
*x^5-693*a^3*b^4*e^7*x^4+594*a^2*b^5*d*e^6*x^4-264*a*b^6*d^2*e^5*x^4+48*b^7*d^3*e^4*x^4-1155*a^4*b^3*e^7*x^3+1
848*a^3*b^4*d*e^6*x^3-1584*a^2*b^5*d^2*e^5*x^3+704*a*b^6*d^3*e^4*x^3-128*b^7*d^4*e^3*x^3-2079*a^5*b^2*e^7*x^2+
6930*a^4*b^3*d*e^6*x^2-11088*a^3*b^4*d^2*e^5*x^2+9504*a^2*b^5*d^3*e^4*x^2-4224*a*b^6*d^4*e^3*x^2+768*b^7*d^5*e
^2*x^2+693*a^6*b*e^7*x-8316*a^5*b^2*d*e^6*x+27720*a^4*b^3*d^2*e^5*x-44352*a^3*b^4*d^3*e^4*x+38016*a^2*b^5*d^4*
e^3*x-16896*a*b^6*d^5*e^2*x+3072*b^7*d^6*e*x+33*a^7*e^7+462*a^6*b*d*e^6-5544*a^5*b^2*d^2*e^5+18480*a^4*b^3*d^3
*e^4-29568*a^3*b^4*d^4*e^3+25344*a^2*b^5*d^5*e^2-11264*a*b^6*d^6*e+2048*b^7*d^7)/(e*x+d)^(3/2)/e^8

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maxima [B]  time = 0.58, size = 462, normalized size = 2.22 \begin {gather*} \frac {2 \, {\left (\frac {9 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{7} - 77 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 297 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 693 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 2079 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \sqrt {e x + d}}{e^{7}} + \frac {33 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7} - 21 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{99 \, 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)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/99*((9*(e*x + d)^(11/2)*b^7 - 77*(b^7*d - a*b^6*e)*(e*x + d)^(9/2) + 297*(b^7*d^2 - 2*a*b^6*d*e + a^2*b^5*e^
2)*(e*x + d)^(7/2) - 693*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(e*x + d)^(5/2) + 1155*(b^7
*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*(e*x + d)^(3/2) - 2079*(b^7*d^5 - 5*
a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 - a^5*b^2*e^5)*sqrt(e*x + d))/e^7 + 33
*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5
+ 7*a^6*b*d*e^6 - a^7*e^7 - 21*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3
*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d))/((e*x + d)^(3/2)*e^7))/e

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mupad [B]  time = 2.05, size = 335, normalized size = 1.61 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}-\frac {\left (d+e\,x\right )\,\left (14\,a^6\,b\,e^6-84\,a^5\,b^2\,d\,e^5+210\,a^4\,b^3\,d^2\,e^4-280\,a^3\,b^4\,d^3\,e^3+210\,a^2\,b^5\,d^4\,e^2-84\,a\,b^6\,d^5\,e+14\,b^7\,d^6\right )+\frac {2\,a^7\,e^7}{3}-\frac {2\,b^7\,d^7}{3}-14\,a^2\,b^5\,d^5\,e^2+\frac {70\,a^3\,b^4\,d^4\,e^3}{3}-\frac {70\,a^4\,b^3\,d^3\,e^4}{3}+14\,a^5\,b^2\,d^2\,e^5+\frac {14\,a\,b^6\,d^6\,e}{3}-\frac {14\,a^6\,b\,d\,e^6}{3}}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,\sqrt {d+e\,x}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {14\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{e^8}+\frac {6\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{e^8} \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)^3)/(d + e*x)^(5/2),x)

[Out]

(2*b^7*(d + e*x)^(11/2))/(11*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(9/2))/(9*e^8) - ((d + e*x)*(14*b^7*d^6
+ 14*a^6*b*e^6 - 84*a^5*b^2*d*e^5 + 210*a^2*b^5*d^4*e^2 - 280*a^3*b^4*d^3*e^3 + 210*a^4*b^3*d^2*e^4 - 84*a*b^
6*d^5*e) + (2*a^7*e^7)/3 - (2*b^7*d^7)/3 - 14*a^2*b^5*d^5*e^2 + (70*a^3*b^4*d^4*e^3)/3 - (70*a^4*b^3*d^3*e^4)/
3 + 14*a^5*b^2*d^2*e^5 + (14*a*b^6*d^6*e)/3 - (14*a^6*b*d*e^6)/3)/(e^8*(d + e*x)^(3/2)) + (42*b^2*(a*e - b*d)^
5*(d + e*x)^(1/2))/e^8 + (70*b^3*(a*e - b*d)^4*(d + e*x)^(3/2))/(3*e^8) + (14*b^4*(a*e - b*d)^3*(d + e*x)^(5/2
))/e^8 + (6*b^5*(a*e - b*d)^2*(d + e*x)^(7/2))/e^8

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sympy [A]  time = 113.69, size = 360, normalized size = 1.73 \begin {gather*} \frac {2 b^{7} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{8}} - \frac {14 b \left (a e - b d\right )^{6}}{e^{8} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{9 e^{8}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{7 e^{8}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (70 a^{3} b^{4} e^{3} - 210 a^{2} b^{5} d e^{2} + 210 a b^{6} d^{2} e - 70 b^{7} d^{3}\right )}{5 e^{8}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (70 a^{4} b^{3} e^{4} - 280 a^{3} b^{4} d e^{3} + 420 a^{2} b^{5} d^{2} e^{2} - 280 a b^{6} d^{3} e + 70 b^{7} d^{4}\right )}{3 e^{8}} + \frac {\sqrt {d + e x} \left (42 a^{5} b^{2} e^{5} - 210 a^{4} b^{3} d e^{4} + 420 a^{3} b^{4} d^{2} e^{3} - 420 a^{2} b^{5} d^{3} e^{2} + 210 a b^{6} d^{4} e - 42 b^{7} d^{5}\right )}{e^{8}} - \frac {2 \left (a e - b d\right )^{7}}{3 e^{8} \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)**3/(e*x+d)**(5/2),x)

[Out]

2*b**7*(d + e*x)**(11/2)/(11*e**8) - 14*b*(a*e - b*d)**6/(e**8*sqrt(d + e*x)) + (d + e*x)**(9/2)*(14*a*b**6*e
- 14*b**7*d)/(9*e**8) + (d + e*x)**(7/2)*(42*a**2*b**5*e**2 - 84*a*b**6*d*e + 42*b**7*d**2)/(7*e**8) + (d + e*
x)**(5/2)*(70*a**3*b**4*e**3 - 210*a**2*b**5*d*e**2 + 210*a*b**6*d**2*e - 70*b**7*d**3)/(5*e**8) + (d + e*x)**
(3/2)*(70*a**4*b**3*e**4 - 280*a**3*b**4*d*e**3 + 420*a**2*b**5*d**2*e**2 - 280*a*b**6*d**3*e + 70*b**7*d**4)/
(3*e**8) + sqrt(d + e*x)*(42*a**5*b**2*e**5 - 210*a**4*b**3*d*e**4 + 420*a**3*b**4*d**2*e**3 - 420*a**2*b**5*d
**3*e**2 + 210*a*b**6*d**4*e - 42*b**7*d**5)/e**8 - 2*(a*e - b*d)**7/(3*e**8*(d + e*x)**(3/2))

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