3.19.36 \(\int (a+b x) \sqrt {d+e x} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=214 \[ -\frac {14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac {42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac {70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac {70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac {6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac {14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8} \]

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Rubi [A]  time = 0.07, 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, 43} \begin {gather*} -\frac {14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac {42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac {70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac {70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac {6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac {14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac {2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8} \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)^3,x]

[Out]

(-2*(b*d - a*e)^7*(d + e*x)^(3/2))/(3*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(5/2))/(5*e^8) - (6*b^2*(b*d - a*e)
^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(11
/2))/(11*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(15/2))/(15*e
^8) + (2*b^7*(d + e*x)^(17/2))/(17*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 (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {(-b d+a e)^7 \sqrt {d+e x}}{e^7}+\frac {7 b (b d-a e)^6 (d+e x)^{3/2}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{5/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{9/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{11/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{13/2}}{e^7}+\frac {b^7 (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 (d+e x)^{3/2}}{3 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{5/2}}{5 e^8}-\frac {6 b^2 (b d-a e)^5 (d+e x)^{7/2}}{e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{9/2}}{9 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{11/2}}{11 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{13/2}}{13 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{15/2}}{15 e^8}+\frac {2 b^7 (d+e x)^{17/2}}{17 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 167, normalized size = 0.78 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-51051 b^6 (d+e x)^6 (b d-a e)+176715 b^5 (d+e x)^5 (b d-a e)^2-348075 b^4 (d+e x)^4 (b d-a e)^3+425425 b^3 (d+e x)^3 (b d-a e)^4-328185 b^2 (d+e x)^2 (b d-a e)^5+153153 b (d+e x) (b d-a e)^6-36465 (b d-a e)^7+6435 b^7 (d+e x)^7\right )}{109395 e^8} \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)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(-36465*(b*d - a*e)^7 + 153153*b*(b*d - a*e)^6*(d + e*x) - 328185*b^2*(b*d - a*e)^5*(d + e*
x)^2 + 425425*b^3*(b*d - a*e)^4*(d + e*x)^3 - 348075*b^4*(b*d - a*e)^3*(d + e*x)^4 + 176715*b^5*(b*d - a*e)^2*
(d + e*x)^5 - 51051*b^6*(b*d - a*e)*(d + e*x)^6 + 6435*b^7*(d + e*x)^7))/(109395*e^8)

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IntegrateAlgebraic [B]  time = 0.17, size = 582, normalized size = 2.72 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (36465 a^7 e^7+153153 a^6 b e^6 (d+e x)-255255 a^6 b d e^6+765765 a^5 b^2 d^2 e^5+328185 a^5 b^2 e^5 (d+e x)^2-918918 a^5 b^2 d e^5 (d+e x)-1276275 a^4 b^3 d^3 e^4+2297295 a^4 b^3 d^2 e^4 (d+e x)+425425 a^4 b^3 e^4 (d+e x)^3-1640925 a^4 b^3 d e^4 (d+e x)^2+1276275 a^3 b^4 d^4 e^3-3063060 a^3 b^4 d^3 e^3 (d+e x)+3281850 a^3 b^4 d^2 e^3 (d+e x)^2+348075 a^3 b^4 e^3 (d+e x)^4-1701700 a^3 b^4 d e^3 (d+e x)^3-765765 a^2 b^5 d^5 e^2+2297295 a^2 b^5 d^4 e^2 (d+e x)-3281850 a^2 b^5 d^3 e^2 (d+e x)^2+2552550 a^2 b^5 d^2 e^2 (d+e x)^3+176715 a^2 b^5 e^2 (d+e x)^5-1044225 a^2 b^5 d e^2 (d+e x)^4+255255 a b^6 d^6 e-918918 a b^6 d^5 e (d+e x)+1640925 a b^6 d^4 e (d+e x)^2-1701700 a b^6 d^3 e (d+e x)^3+1044225 a b^6 d^2 e (d+e x)^4+51051 a b^6 e (d+e x)^6-353430 a b^6 d e (d+e x)^5-36465 b^7 d^7+153153 b^7 d^6 (d+e x)-328185 b^7 d^5 (d+e x)^2+425425 b^7 d^4 (d+e x)^3-348075 b^7 d^3 (d+e x)^4+176715 b^7 d^2 (d+e x)^5+6435 b^7 (d+e x)^7-51051 b^7 d (d+e x)^6\right )}{109395 e^8} \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)^3,x]

[Out]

(2*(d + e*x)^(3/2)*(-36465*b^7*d^7 + 255255*a*b^6*d^6*e - 765765*a^2*b^5*d^5*e^2 + 1276275*a^3*b^4*d^4*e^3 - 1
276275*a^4*b^3*d^3*e^4 + 765765*a^5*b^2*d^2*e^5 - 255255*a^6*b*d*e^6 + 36465*a^7*e^7 + 153153*b^7*d^6*(d + e*x
) - 918918*a*b^6*d^5*e*(d + e*x) + 2297295*a^2*b^5*d^4*e^2*(d + e*x) - 3063060*a^3*b^4*d^3*e^3*(d + e*x) + 229
7295*a^4*b^3*d^2*e^4*(d + e*x) - 918918*a^5*b^2*d*e^5*(d + e*x) + 153153*a^6*b*e^6*(d + e*x) - 328185*b^7*d^5*
(d + e*x)^2 + 1640925*a*b^6*d^4*e*(d + e*x)^2 - 3281850*a^2*b^5*d^3*e^2*(d + e*x)^2 + 3281850*a^3*b^4*d^2*e^3*
(d + e*x)^2 - 1640925*a^4*b^3*d*e^4*(d + e*x)^2 + 328185*a^5*b^2*e^5*(d + e*x)^2 + 425425*b^7*d^4*(d + e*x)^3
- 1701700*a*b^6*d^3*e*(d + e*x)^3 + 2552550*a^2*b^5*d^2*e^2*(d + e*x)^3 - 1701700*a^3*b^4*d*e^3*(d + e*x)^3 +
425425*a^4*b^3*e^4*(d + e*x)^3 - 348075*b^7*d^3*(d + e*x)^4 + 1044225*a*b^6*d^2*e*(d + e*x)^4 - 1044225*a^2*b^
5*d*e^2*(d + e*x)^4 + 348075*a^3*b^4*e^3*(d + e*x)^4 + 176715*b^7*d^2*(d + e*x)^5 - 353430*a*b^6*d*e*(d + e*x)
^5 + 176715*a^2*b^5*e^2*(d + e*x)^5 - 51051*b^7*d*(d + e*x)^6 + 51051*a*b^6*e*(d + e*x)^6 + 6435*b^7*(d + e*x)
^7))/(109395*e^8)

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fricas [B]  time = 0.43, size = 568, normalized size = 2.65 \begin {gather*} \frac {2 \, {\left (6435 \, b^{7} e^{8} x^{8} - 2048 \, b^{7} d^{8} + 17408 \, a b^{6} d^{7} e - 65280 \, a^{2} b^{5} d^{6} e^{2} + 141440 \, a^{3} b^{4} d^{5} e^{3} - 194480 \, a^{4} b^{3} d^{4} e^{4} + 175032 \, a^{5} b^{2} d^{3} e^{5} - 102102 \, a^{6} b d^{2} e^{6} + 36465 \, a^{7} d e^{7} + 429 \, {\left (b^{7} d e^{7} + 119 \, a b^{6} e^{8}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{2} e^{6} - 17 \, a b^{6} d e^{7} - 765 \, a^{2} b^{5} e^{8}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{3} e^{5} - 68 \, a b^{6} d^{2} e^{6} + 255 \, a^{2} b^{5} d e^{7} + 5525 \, a^{3} b^{4} e^{8}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{4} e^{4} - 136 \, a b^{6} d^{3} e^{5} + 510 \, a^{2} b^{5} d^{2} e^{6} - 1105 \, a^{3} b^{4} d e^{7} - 12155 \, a^{4} b^{3} e^{8}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{5} e^{3} - 1088 \, a b^{6} d^{4} e^{4} + 4080 \, a^{2} b^{5} d^{3} e^{5} - 8840 \, a^{3} b^{4} d^{2} e^{6} + 12155 \, a^{4} b^{3} d e^{7} + 65637 \, a^{5} b^{2} e^{8}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{6} e^{2} - 2176 \, a b^{6} d^{5} e^{3} + 8160 \, a^{2} b^{5} d^{4} e^{4} - 17680 \, a^{3} b^{4} d^{3} e^{5} + 24310 \, a^{4} b^{3} d^{2} e^{6} - 21879 \, a^{5} b^{2} d e^{7} - 51051 \, a^{6} b e^{8}\right )} x^{2} + {\left (1024 \, b^{7} d^{7} e - 8704 \, a b^{6} d^{6} e^{2} + 32640 \, a^{2} b^{5} d^{5} e^{3} - 70720 \, a^{3} b^{4} d^{4} e^{4} + 97240 \, a^{4} b^{3} d^{3} e^{5} - 87516 \, a^{5} b^{2} d^{2} e^{6} + 51051 \, a^{6} b d e^{7} + 36465 \, a^{7} e^{8}\right )} x\right )} \sqrt {e x + d}}{109395 \, e^{8}} \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)^(1/2),x, algorithm="fricas")

[Out]

2/109395*(6435*b^7*e^8*x^8 - 2048*b^7*d^8 + 17408*a*b^6*d^7*e - 65280*a^2*b^5*d^6*e^2 + 141440*a^3*b^4*d^5*e^3
- 194480*a^4*b^3*d^4*e^4 + 175032*a^5*b^2*d^3*e^5 - 102102*a^6*b*d^2*e^6 + 36465*a^7*d*e^7 + 429*(b^7*d*e^7 +
119*a*b^6*e^8)*x^7 - 231*(2*b^7*d^2*e^6 - 17*a*b^6*d*e^7 - 765*a^2*b^5*e^8)*x^6 + 63*(8*b^7*d^3*e^5 - 68*a*b^
6*d^2*e^6 + 255*a^2*b^5*d*e^7 + 5525*a^3*b^4*e^8)*x^5 - 35*(16*b^7*d^4*e^4 - 136*a*b^6*d^3*e^5 + 510*a^2*b^5*d
^2*e^6 - 1105*a^3*b^4*d*e^7 - 12155*a^4*b^3*e^8)*x^4 + 5*(128*b^7*d^5*e^3 - 1088*a*b^6*d^4*e^4 + 4080*a^2*b^5*
d^3*e^5 - 8840*a^3*b^4*d^2*e^6 + 12155*a^4*b^3*d*e^7 + 65637*a^5*b^2*e^8)*x^3 - 3*(256*b^7*d^6*e^2 - 2176*a*b^
6*d^5*e^3 + 8160*a^2*b^5*d^4*e^4 - 17680*a^3*b^4*d^3*e^5 + 24310*a^4*b^3*d^2*e^6 - 21879*a^5*b^2*d*e^7 - 51051
*a^6*b*e^8)*x^2 + (1024*b^7*d^7*e - 8704*a*b^6*d^6*e^2 + 32640*a^2*b^5*d^5*e^3 - 70720*a^3*b^4*d^4*e^4 + 97240
*a^4*b^3*d^3*e^5 - 87516*a^5*b^2*d^2*e^6 + 51051*a^6*b*d*e^7 + 36465*a^7*e^8)*x)*sqrt(e*x + d)/e^8

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giac [B]  time = 0.22, size = 1119, normalized size = 5.23

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)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/109395*(255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*b*d*e^(-1) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e +
d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b^2*d*e^(-2) + 109395*(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^4*b^3*d*e^(-3) + 12155*(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^3*b^4*d*e^(-4) + 3315*(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^2*b^5*d*e^(-5) + 255*(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*sq
rt(x*e + d)*d^6)*a*b^6*d*e^(-6) + 17*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*
d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2
)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^7*d*e^(-7) + 51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*a^6*b*e^(-1) + 65637*(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^5*b^2*e^(-2) + 12155*(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^4*b^3*e^(-3) + 5525*(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^3
*b^4*e^(-4) + 765*(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)*a^2*b^5*e^(-5) + 119
*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32
175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a*b^
6*e^(-6) + (6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^
(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*
e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*b^7*e^(-7) + 109395*sqrt(x*e + d)*a^7*d + 36465*((x*e + d)^(3/2)
- 3*sqrt(x*e + d)*d)*a^7)*e^(-1)

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maple [B]  time = 0.05, size = 498, normalized size = 2.33 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6435 b^{7} x^{7} e^{7}+51051 a \,b^{6} e^{7} x^{6}-6006 b^{7} d \,e^{6} x^{6}+176715 a^{2} b^{5} e^{7} x^{5}-47124 a \,b^{6} d \,e^{6} x^{5}+5544 b^{7} d^{2} e^{5} x^{5}+348075 a^{3} b^{4} e^{7} x^{4}-160650 a^{2} b^{5} d \,e^{6} x^{4}+42840 a \,b^{6} d^{2} e^{5} x^{4}-5040 b^{7} d^{3} e^{4} x^{4}+425425 a^{4} b^{3} e^{7} x^{3}-309400 a^{3} b^{4} d \,e^{6} x^{3}+142800 a^{2} b^{5} d^{2} e^{5} x^{3}-38080 a \,b^{6} d^{3} e^{4} x^{3}+4480 b^{7} d^{4} e^{3} x^{3}+328185 a^{5} b^{2} e^{7} x^{2}-364650 a^{4} b^{3} d \,e^{6} x^{2}+265200 a^{3} b^{4} d^{2} e^{5} x^{2}-122400 a^{2} b^{5} d^{3} e^{4} x^{2}+32640 a \,b^{6} d^{4} e^{3} x^{2}-3840 b^{7} d^{5} e^{2} x^{2}+153153 a^{6} b \,e^{7} x -262548 a^{5} b^{2} d \,e^{6} x +291720 a^{4} b^{3} d^{2} e^{5} x -212160 a^{3} b^{4} d^{3} e^{4} x +97920 a^{2} b^{5} d^{4} e^{3} x -26112 a \,b^{6} d^{5} e^{2} x +3072 b^{7} d^{6} e x +36465 a^{7} e^{7}-102102 a^{6} b d \,e^{6}+175032 a^{5} b^{2} d^{2} e^{5}-194480 a^{4} b^{3} d^{3} e^{4}+141440 a^{3} b^{4} d^{4} e^{3}-65280 a^{2} b^{5} d^{5} e^{2}+17408 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{109395 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)^(1/2),x)

[Out]

2/109395*(e*x+d)^(3/2)*(6435*b^7*e^7*x^7+51051*a*b^6*e^7*x^6-6006*b^7*d*e^6*x^6+176715*a^2*b^5*e^7*x^5-47124*a
*b^6*d*e^6*x^5+5544*b^7*d^2*e^5*x^5+348075*a^3*b^4*e^7*x^4-160650*a^2*b^5*d*e^6*x^4+42840*a*b^6*d^2*e^5*x^4-50
40*b^7*d^3*e^4*x^4+425425*a^4*b^3*e^7*x^3-309400*a^3*b^4*d*e^6*x^3+142800*a^2*b^5*d^2*e^5*x^3-38080*a*b^6*d^3*
e^4*x^3+4480*b^7*d^4*e^3*x^3+328185*a^5*b^2*e^7*x^2-364650*a^4*b^3*d*e^6*x^2+265200*a^3*b^4*d^2*e^5*x^2-122400
*a^2*b^5*d^3*e^4*x^2+32640*a*b^6*d^4*e^3*x^2-3840*b^7*d^5*e^2*x^2+153153*a^6*b*e^7*x-262548*a^5*b^2*d*e^6*x+29
1720*a^4*b^3*d^2*e^5*x-212160*a^3*b^4*d^3*e^4*x+97920*a^2*b^5*d^4*e^3*x-26112*a*b^6*d^5*e^2*x+3072*b^7*d^6*e*x
+36465*a^7*e^7-102102*a^6*b*d*e^6+175032*a^5*b^2*d^2*e^5-194480*a^4*b^3*d^3*e^4+141440*a^3*b^4*d^4*e^3-65280*a
^2*b^5*d^5*e^2+17408*a*b^6*d^6*e-2048*b^7*d^7)/e^8

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maxima [B]  time = 0.60, size = 456, normalized size = 2.13 \begin {gather*} \frac {2 \, {\left (6435 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{7} - 51051 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 348075 \, {\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 {11}{2}} + 425425 \, {\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 {9}{2}} - 328185 \, {\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 )} {\left (e x + d\right )}^{\frac {7}{2}} + 153153 \, {\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 )}^{\frac {5}{2}} - 36465 \, {\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}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{109395 \, e^{8}} \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)^(1/2),x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*b^7 - 51051*(b^7*d - a*b^6*e)*(e*x + d)^(15/2) + 176715*(b^7*d^2 - 2*a*b^6*d*e
+ a^2*b^5*e^2)*(e*x + d)^(13/2) - 348075*(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)^
(11/2) + 425425*(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)^(9/2)
- 328185*(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)*(
e*x + d)^(7/2) + 153153*(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)^(5/2) - 36465*(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)*(e*x + d)^(3/2))/e^8

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mupad [B]  time = 2.05, size = 187, normalized size = 0.87 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8}+\frac {6\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8} \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)^3,x)

[Out]

(2*b^7*(d + e*x)^(17/2))/(17*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(15/2))/(15*e^8) + (2*(a*e - b*d)^7*(d
+ e*x)^(3/2))/(3*e^8) + (6*b^2*(a*e - b*d)^5*(d + e*x)^(7/2))/e^8 + (70*b^3*(a*e - b*d)^4*(d + e*x)^(9/2))/(9*
e^8) + (70*b^4*(a*e - b*d)^3*(d + e*x)^(11/2))/(11*e^8) + (42*b^5*(a*e - b*d)^2*(d + e*x)^(13/2))/(13*e^8) + (
14*b*(a*e - b*d)^6*(d + e*x)^(5/2))/(5*e^8)

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sympy [B]  time = 10.16, size = 544, normalized size = 2.54 \begin {gather*} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (7 a b^{6} e - 7 b^{7} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{3 e^{7}}\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)**3*(e*x+d)**(1/2),x)

[Out]

2*(b**7*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(7*a*b**6*e - 7*b**7*d)/(15*e**7) + (d + e*x)**(13/2)*
(21*a**2*b**5*e**2 - 42*a*b**6*d*e + 21*b**7*d**2)/(13*e**7) + (d + e*x)**(11/2)*(35*a**3*b**4*e**3 - 105*a**2
*b**5*d*e**2 + 105*a*b**6*d**2*e - 35*b**7*d**3)/(11*e**7) + (d + e*x)**(9/2)*(35*a**4*b**3*e**4 - 140*a**3*b*
*4*d*e**3 + 210*a**2*b**5*d**2*e**2 - 140*a*b**6*d**3*e + 35*b**7*d**4)/(9*e**7) + (d + e*x)**(7/2)*(21*a**5*b
**2*e**5 - 105*a**4*b**3*d*e**4 + 210*a**3*b**4*d**2*e**3 - 210*a**2*b**5*d**3*e**2 + 105*a*b**6*d**4*e - 21*b
**7*d**5)/(7*e**7) + (d + e*x)**(5/2)*(7*a**6*b*e**6 - 42*a**5*b**2*d*e**5 + 105*a**4*b**3*d**2*e**4 - 140*a**
3*b**4*d**3*e**3 + 105*a**2*b**5*d**4*e**2 - 42*a*b**6*d**5*e + 7*b**7*d**6)/(5*e**7) + (d + e*x)**(3/2)*(a**7
*e**7 - 7*a**6*b*d*e**6 + 21*a**5*b**2*d**2*e**5 - 35*a**4*b**3*d**3*e**4 + 35*a**3*b**4*d**4*e**3 - 21*a**2*b
**5*d**5*e**2 + 7*a*b**6*d**6*e - b**7*d**7)/(3*e**7))/e

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