3.2060 \(\int (a+b x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=214 \[ -\frac{14 b^6 (d+e x)^{17/2} (b d-a e)}{17 e^8}+\frac{14 b^5 (d+e x)^{15/2} (b d-a e)^2}{5 e^8}-\frac{70 b^4 (d+e x)^{13/2} (b d-a e)^3}{13 e^8}+\frac{70 b^3 (d+e x)^{11/2} (b d-a e)^4}{11 e^8}-\frac{14 b^2 (d+e x)^{9/2} (b d-a e)^5}{3 e^8}+\frac{2 b (d+e x)^{7/2} (b d-a e)^6}{e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^7}{5 e^8}+\frac{2 b^7 (d+e x)^{19/2}}{19 e^8} \]

[Out]

(-2*(b*d - a*e)^7*(d + e*x)^(5/2))/(5*e^8) + (2*b*(b*d - a*e)^6*(d + e*x)^(7/2))/e^8 - (14*b^2*(b*d - a*e)^5*(
d + e*x)^(9/2))/(3*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(
13/2))/(13*e^8) + (14*b^5*(b*d - a*e)^2*(d + e*x)^(15/2))/(5*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(17/2))/(17*
e^8) + (2*b^7*(d + e*x)^(19/2))/(19*e^8)

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Rubi [A]  time = 0.0745669, antiderivative size = 214, normalized size of antiderivative = 1., 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} \[ -\frac{14 b^6 (d+e x)^{17/2} (b d-a e)}{17 e^8}+\frac{14 b^5 (d+e x)^{15/2} (b d-a e)^2}{5 e^8}-\frac{70 b^4 (d+e x)^{13/2} (b d-a e)^3}{13 e^8}+\frac{70 b^3 (d+e x)^{11/2} (b d-a e)^4}{11 e^8}-\frac{14 b^2 (d+e x)^{9/2} (b d-a e)^5}{3 e^8}+\frac{2 b (d+e x)^{7/2} (b d-a e)^6}{e^8}-\frac{2 (d+e x)^{5/2} (b d-a e)^7}{5 e^8}+\frac{2 b^7 (d+e x)^{19/2}}{19 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^7*(d + e*x)^(5/2))/(5*e^8) + (2*b*(b*d - a*e)^6*(d + e*x)^(7/2))/e^8 - (14*b^2*(b*d - a*e)^5*(
d + e*x)^(9/2))/(3*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(11/2))/(11*e^8) - (70*b^4*(b*d - a*e)^3*(d + e*x)^(
13/2))/(13*e^8) + (14*b^5*(b*d - a*e)^2*(d + e*x)^(15/2))/(5*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(17/2))/(17*
e^8) + (2*b^7*(d + e*x)^(19/2))/(19*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) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^7 (d+e x)^{3/2}}{e^7}+\frac{7 b (b d-a e)^6 (d+e x)^{5/2}}{e^7}-\frac{21 b^2 (b d-a e)^5 (d+e x)^{7/2}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{9/2}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{11/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{13/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{15/2}}{e^7}+\frac{b^7 (d+e x)^{17/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (b d-a e)^7 (d+e x)^{5/2}}{5 e^8}+\frac{2 b (b d-a e)^6 (d+e x)^{7/2}}{e^8}-\frac{14 b^2 (b d-a e)^5 (d+e x)^{9/2}}{3 e^8}+\frac{70 b^3 (b d-a e)^4 (d+e x)^{11/2}}{11 e^8}-\frac{70 b^4 (b d-a e)^3 (d+e x)^{13/2}}{13 e^8}+\frac{14 b^5 (b d-a e)^2 (d+e x)^{15/2}}{5 e^8}-\frac{14 b^6 (b d-a e) (d+e x)^{17/2}}{17 e^8}+\frac{2 b^7 (d+e x)^{19/2}}{19 e^8}\\ \end{align*}

Mathematica [A]  time = 0.135208, size = 167, normalized size = 0.78 \[ \frac{2 (d+e x)^{5/2} \left (-1616615 b^2 (d+e x)^2 (b d-a e)^5+2204475 b^3 (d+e x)^3 (b d-a e)^4-1865325 b^4 (d+e x)^4 (b d-a e)^3+969969 b^5 (d+e x)^5 (b d-a e)^2-285285 b^6 (d+e x)^6 (b d-a e)+692835 b (d+e x) (b d-a e)^6-138567 (b d-a e)^7+36465 b^7 (d+e x)^7\right )}{692835 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(-138567*(b*d - a*e)^7 + 692835*b*(b*d - a*e)^6*(d + e*x) - 1616615*b^2*(b*d - a*e)^5*(d +
e*x)^2 + 2204475*b^3*(b*d - a*e)^4*(d + e*x)^3 - 1865325*b^4*(b*d - a*e)^3*(d + e*x)^4 + 969969*b^5*(b*d - a*e
)^2*(d + e*x)^5 - 285285*b^6*(b*d - a*e)*(d + e*x)^6 + 36465*b^7*(d + e*x)^7))/(692835*e^8)

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Maple [B]  time = 0.007, size = 498, normalized size = 2.3 \begin{align*}{\frac{72930\,{b}^{7}{x}^{7}{e}^{7}+570570\,a{b}^{6}{e}^{7}{x}^{6}-60060\,{b}^{7}d{e}^{6}{x}^{6}+1939938\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-456456\,a{b}^{6}d{e}^{6}{x}^{5}+48048\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+3730650\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-1492260\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+351120\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-36960\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+4408950\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-2713200\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+1085280\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-255360\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+26880\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+3233230\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-2939300\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+1808800\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-723520\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+170240\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-17920\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+1385670\,{a}^{6}b{e}^{7}x-1847560\,{a}^{5}{b}^{2}d{e}^{6}x+1679600\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-1033600\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+413440\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-97280\,a{b}^{6}{d}^{5}{e}^{2}x+10240\,{b}^{7}{d}^{6}ex+277134\,{a}^{7}{e}^{7}-554268\,{a}^{6}bd{e}^{6}+739024\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-671840\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+413440\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-165376\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+38912\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{692835\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/692835*(e*x+d)^(5/2)*(36465*b^7*e^7*x^7+285285*a*b^6*e^7*x^6-30030*b^7*d*e^6*x^6+969969*a^2*b^5*e^7*x^5-2282
28*a*b^6*d*e^6*x^5+24024*b^7*d^2*e^5*x^5+1865325*a^3*b^4*e^7*x^4-746130*a^2*b^5*d*e^6*x^4+175560*a*b^6*d^2*e^5
*x^4-18480*b^7*d^3*e^4*x^4+2204475*a^4*b^3*e^7*x^3-1356600*a^3*b^4*d*e^6*x^3+542640*a^2*b^5*d^2*e^5*x^3-127680
*a*b^6*d^3*e^4*x^3+13440*b^7*d^4*e^3*x^3+1616615*a^5*b^2*e^7*x^2-1469650*a^4*b^3*d*e^6*x^2+904400*a^3*b^4*d^2*
e^5*x^2-361760*a^2*b^5*d^3*e^4*x^2+85120*a*b^6*d^4*e^3*x^2-8960*b^7*d^5*e^2*x^2+692835*a^6*b*e^7*x-923780*a^5*
b^2*d*e^6*x+839800*a^4*b^3*d^2*e^5*x-516800*a^3*b^4*d^3*e^4*x+206720*a^2*b^5*d^4*e^3*x-48640*a*b^6*d^5*e^2*x+5
120*b^7*d^6*e*x+138567*a^7*e^7-277134*a^6*b*d*e^6+369512*a^5*b^2*d^2*e^5-335920*a^4*b^3*d^3*e^4+206720*a^3*b^4
*d^4*e^3-82688*a^2*b^5*d^5*e^2+19456*a*b^6*d^6*e-2048*b^7*d^7)/e^8

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Maxima [B]  time = 0.979521, size = 616, normalized size = 2.88 \begin{align*} \frac{2 \,{\left (36465 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{7} - 285285 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1865325 \,{\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{13}{2}} + 2204475 \,{\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{11}{2}} - 1616615 \,{\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{9}{2}} + 692835 \,{\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{7}{2}} - 138567 \,{\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{5}{2}}\right )}}{692835 \, e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/692835*(36465*(e*x + d)^(19/2)*b^7 - 285285*(b^7*d - a*b^6*e)*(e*x + d)^(17/2) + 969969*(b^7*d^2 - 2*a*b^6*d
*e + a^2*b^5*e^2)*(e*x + d)^(15/2) - 1865325*(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)^(13/2) + 2204475*(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)^(1
1/2) - 1616615*(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)^(9/2) + 692835*(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)^(7/2) - 138567*(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)^(5/2))/e^8

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Fricas [B]  time = 1.38877, size = 1596, normalized size = 7.46 \begin{align*} \frac{2 \,{\left (36465 \, b^{7} e^{9} x^{9} - 2048 \, b^{7} d^{9} + 19456 \, a b^{6} d^{8} e - 82688 \, a^{2} b^{5} d^{7} e^{2} + 206720 \, a^{3} b^{4} d^{6} e^{3} - 335920 \, a^{4} b^{3} d^{5} e^{4} + 369512 \, a^{5} b^{2} d^{4} e^{5} - 277134 \, a^{6} b d^{3} e^{6} + 138567 \, a^{7} d^{2} e^{7} + 2145 \,{\left (20 \, b^{7} d e^{8} + 133 \, a b^{6} e^{9}\right )} x^{8} + 429 \,{\left (b^{7} d^{2} e^{7} + 798 \, a b^{6} d e^{8} + 2261 \, a^{2} b^{5} e^{9}\right )} x^{7} - 231 \,{\left (2 \, b^{7} d^{3} e^{6} - 19 \, a b^{6} d^{2} e^{7} - 5168 \, a^{2} b^{5} d e^{8} - 8075 \, a^{3} b^{4} e^{9}\right )} x^{6} + 21 \,{\left (24 \, b^{7} d^{4} e^{5} - 228 \, a b^{6} d^{3} e^{6} + 969 \, a^{2} b^{5} d^{2} e^{7} + 113050 \, a^{3} b^{4} d e^{8} + 104975 \, a^{4} b^{3} e^{9}\right )} x^{5} - 35 \,{\left (16 \, b^{7} d^{5} e^{4} - 152 \, a b^{6} d^{4} e^{5} + 646 \, a^{2} b^{5} d^{3} e^{6} - 1615 \, a^{3} b^{4} d^{2} e^{7} - 83980 \, a^{4} b^{3} d e^{8} - 46189 \, a^{5} b^{2} e^{9}\right )} x^{4} + 5 \,{\left (128 \, b^{7} d^{6} e^{3} - 1216 \, a b^{6} d^{5} e^{4} + 5168 \, a^{2} b^{5} d^{4} e^{5} - 12920 \, a^{3} b^{4} d^{3} e^{6} + 20995 \, a^{4} b^{3} d^{2} e^{7} + 461890 \, a^{5} b^{2} d e^{8} + 138567 \, a^{6} b e^{9}\right )} x^{3} - 3 \,{\left (256 \, b^{7} d^{7} e^{2} - 2432 \, a b^{6} d^{6} e^{3} + 10336 \, a^{2} b^{5} d^{5} e^{4} - 25840 \, a^{3} b^{4} d^{4} e^{5} + 41990 \, a^{4} b^{3} d^{3} e^{6} - 46189 \, a^{5} b^{2} d^{2} e^{7} - 369512 \, a^{6} b d e^{8} - 46189 \, a^{7} e^{9}\right )} x^{2} +{\left (1024 \, b^{7} d^{8} e - 9728 \, a b^{6} d^{7} e^{2} + 41344 \, a^{2} b^{5} d^{6} e^{3} - 103360 \, a^{3} b^{4} d^{5} e^{4} + 167960 \, a^{4} b^{3} d^{4} e^{5} - 184756 \, a^{5} b^{2} d^{3} e^{6} + 138567 \, a^{6} b d^{2} e^{7} + 277134 \, a^{7} d e^{8}\right )} x\right )} \sqrt{e x + d}}{692835 \, e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/692835*(36465*b^7*e^9*x^9 - 2048*b^7*d^9 + 19456*a*b^6*d^8*e - 82688*a^2*b^5*d^7*e^2 + 206720*a^3*b^4*d^6*e^
3 - 335920*a^4*b^3*d^5*e^4 + 369512*a^5*b^2*d^4*e^5 - 277134*a^6*b*d^3*e^6 + 138567*a^7*d^2*e^7 + 2145*(20*b^7
*d*e^8 + 133*a*b^6*e^9)*x^8 + 429*(b^7*d^2*e^7 + 798*a*b^6*d*e^8 + 2261*a^2*b^5*e^9)*x^7 - 231*(2*b^7*d^3*e^6
- 19*a*b^6*d^2*e^7 - 5168*a^2*b^5*d*e^8 - 8075*a^3*b^4*e^9)*x^6 + 21*(24*b^7*d^4*e^5 - 228*a*b^6*d^3*e^6 + 969
*a^2*b^5*d^2*e^7 + 113050*a^3*b^4*d*e^8 + 104975*a^4*b^3*e^9)*x^5 - 35*(16*b^7*d^5*e^4 - 152*a*b^6*d^4*e^5 + 6
46*a^2*b^5*d^3*e^6 - 1615*a^3*b^4*d^2*e^7 - 83980*a^4*b^3*d*e^8 - 46189*a^5*b^2*e^9)*x^4 + 5*(128*b^7*d^6*e^3
- 1216*a*b^6*d^5*e^4 + 5168*a^2*b^5*d^4*e^5 - 12920*a^3*b^4*d^3*e^6 + 20995*a^4*b^3*d^2*e^7 + 461890*a^5*b^2*d
*e^8 + 138567*a^6*b*e^9)*x^3 - 3*(256*b^7*d^7*e^2 - 2432*a*b^6*d^6*e^3 + 10336*a^2*b^5*d^5*e^4 - 25840*a^3*b^4
*d^4*e^5 + 41990*a^4*b^3*d^3*e^6 - 46189*a^5*b^2*d^2*e^7 - 369512*a^6*b*d*e^8 - 46189*a^7*e^9)*x^2 + (1024*b^7
*d^8*e - 9728*a*b^6*d^7*e^2 + 41344*a^2*b^5*d^6*e^3 - 103360*a^3*b^4*d^5*e^4 + 167960*a^4*b^3*d^4*e^5 - 184756
*a^5*b^2*d^3*e^6 + 138567*a^6*b*d^2*e^7 + 277134*a^7*d*e^8)*x)*sqrt(e*x + d)/e^8

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Sympy [A]  time = 35.755, size = 1265, normalized size = 5.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**7*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**7*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 14*a**6*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 14*a**6*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 42*a**5*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 42*a**5*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5
/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 70*a**4*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*
(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 70*a**4*b**3*(d**4*(d + e*x)**(3/2)/3
- 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4
+ 70*a**3*b**4*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d +
e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 70*a**3*b**4*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) -
10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
5 + 42*a**2*b**5*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d
+ e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 42*a**2*b**5*(d**6*(d + e*x)**(3/2)
/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**
(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 14*a*b**6*d*(d**6*(d + e*x)**(3/2)/3 - 6*d
**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/1
1 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 14*a*b**6*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d +
e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*
(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 2*b**7*d*(-d**7*(d + e*x)**(3/2
)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(1
1/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 2*b**7*(d**8*
(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**
4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/1
7 + (d + e*x)**(19/2)/19)/e**8

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Giac [B]  time = 1.21746, size = 1516, normalized size = 7.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2078505*(969969*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*b*d*e^(-1) + 415701*(15*(x*e + d)^(7/2) - 42*(
x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b^2*d*e^(-2) + 230945*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)
*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^4*b^3*d*e^(-3) + 20995*(315*(x*e + d)^(11/2) - 1540*
(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^3*b^4*d*
e^(-4) + 4845*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7
/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^2*b^5*d*e^(-5) + 323*(3003*(x*e + d)^(15/2) -
20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^
4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*b^6*d*e^(-6) + 19*(6435*(x*e + d)^(17/2) - 51051*
(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 -
328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e + d)^(3/2)*d^7)*b^7*d*e^(-7) + 692835*(x*
e + d)^(3/2)*a^7*d + 138567*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^6*b*e^(-1)
+ 138567*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^5*
b^2*e^(-2) + 20995*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^
(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a^4*b^3*e^(-3) + 8075*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a
^3*b^4*e^(-4) + 969*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x
*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a^2*b^5
*e^(-5) + 133*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e +
d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(
x*e + d)^(3/2)*d^7)*a*b^6*e^(-6) + (109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(1
5/2)*d^2 - 8953560*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 83140
20*(x*e + d)^(7/2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*b^7*e^(-7) + 138567*(3*(x*e
+ d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^7)*e^(-1)