3.1611 \(\int \frac{(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=421 \[ \frac{2 (d+e x)^{5/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac{10 c (d+e x)^{7/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 \sqrt{d+e x}}-\frac{14 c^3 (d+e x)^{11/2} (2 c d-b e)}{11 e^8}+\frac{4 c^4 (d+e x)^{13/2}}{13 e^8} \]

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(7*b*d - a*e))*Sqrt[d + e*x])/e^8 - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2
*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(3/2))/e^8 + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20
*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(5/2))/(5*e^8) - (10*c*(
2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(14*c^2*d^2 + 3*b^2
*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(9/2))/(3*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^8) + (4*c
^4*(d + e*x)^(13/2))/(13*e^8)

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Rubi [A]  time = 0.238615, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {771} \[ \frac{2 (d+e x)^{5/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{5 e^8}+\frac{2 c^2 (d+e x)^{9/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac{10 c (d+e x)^{7/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{2 \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 \sqrt{d+e x}}-\frac{14 c^3 (d+e x)^{11/2} (2 c d-b e)}{11 e^8}+\frac{4 c^4 (d+e x)^{13/2}}{13 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(7*b*d - a*e))*Sqrt[d + e*x])/e^8 - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2
*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(3/2))/e^8 + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20
*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(5/2))/(5*e^8) - (10*c*(
2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(14*c^2*d^2 + 3*b^2
*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(9/2))/(3*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^8) + (4*c
^4*(d + e*x)^(13/2))/(13*e^8)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^{3/2}}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^7 \sqrt{d+e x}}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) \sqrt{d+e x}}{e^7}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3/2}}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{9/2}}{e^7}+\frac{2 c^4 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 \sqrt{d+e x}}+\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \sqrt{d+e x}}{e^8}-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{3/2}}{e^8}+\frac{2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{5/2}}{5 e^8}-\frac{10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^8}+\frac{2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{9/2}}{3 e^8}-\frac{14 c^3 (2 c d-b e) (d+e x)^{11/2}}{11 e^8}+\frac{4 c^4 (d+e x)^{13/2}}{13 e^8}\\ \end{align*}

Mathematica [A]  time = 0.554232, size = 594, normalized size = 1.41 \[ \frac{2 \left (429 c^2 e^2 \left (42 a^2 e^2 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+15 a b e \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )+5 b^2 \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )+429 c e^3 \left (105 a^2 b e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+70 a^3 e^3 (2 d+e x)+84 a b^2 e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )-5 b^3 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )+3003 b e^4 \left (15 a^2 b e^2 (2 d+e x)-5 a^3 e^3+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )-65 c^3 e \left (7 b \left (-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+512 d^5 e x+1024 d^6+28 d e^5 x^5-21 e^6 x^6\right )-22 a e \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )+70 c^4 \left (-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5+1024 d^6 e x+2048 d^7-42 d e^6 x^6+33 e^7 x^7\right )\right )}{15015 e^8 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(70*c^4*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56*d^2*e^5*x^5 - 42
*d*e^6*x^6 + 33*e^7*x^7) + 3003*b*e^4*(-5*a^3*e^3 + 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e
^2*x^2) + b^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) + 429*c*e^3*(70*a^3*e^3*(2*d + e*x) + 105*a^2*b*e^
2*(-8*d^2 - 4*d*e*x + e^2*x^2) + 84*a*b^2*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) - 5*b^3*(128*d^4 + 64
*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 429*c^2*e^2*(42*a^2*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*
x^2 + e^3*x^3) + 15*a*b*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*b^2*(256*d^5
+ 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)) - 65*c^3*e*(-22*a*e*(256*d^5 + 12
8*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) + 7*b*(1024*d^6 + 512*d^5*e*x - 128*d^
4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6))))/(15015*e^8*Sqrt[d + e*x])

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Maple [B]  time = 0.008, size = 795, normalized size = 1.9 \begin{align*} -{\frac{-4620\,{c}^{4}{x}^{7}{e}^{7}-19110\,b{c}^{3}{e}^{7}{x}^{6}+5880\,{c}^{4}d{e}^{6}{x}^{6}-20020\,a{c}^{3}{e}^{7}{x}^{5}-30030\,{b}^{2}{c}^{2}{e}^{7}{x}^{5}+25480\,b{c}^{3}d{e}^{6}{x}^{5}-7840\,{c}^{4}{d}^{2}{e}^{5}{x}^{5}-64350\,ab{c}^{2}{e}^{7}{x}^{4}+28600\,a{c}^{3}d{e}^{6}{x}^{4}-21450\,{b}^{3}c{e}^{7}{x}^{4}+42900\,{b}^{2}{c}^{2}d{e}^{6}{x}^{4}-36400\,b{c}^{3}{d}^{2}{e}^{5}{x}^{4}+11200\,{c}^{4}{d}^{3}{e}^{4}{x}^{4}-36036\,{a}^{2}{c}^{2}{e}^{7}{x}^{3}-72072\,a{b}^{2}c{e}^{7}{x}^{3}+102960\,ab{c}^{2}d{e}^{6}{x}^{3}-45760\,a{c}^{3}{d}^{2}{e}^{5}{x}^{3}-6006\,{b}^{4}{e}^{7}{x}^{3}+34320\,{b}^{3}cd{e}^{6}{x}^{3}-68640\,{b}^{2}{c}^{2}{d}^{2}{e}^{5}{x}^{3}+58240\,b{c}^{3}{d}^{3}{e}^{4}{x}^{3}-17920\,{c}^{4}{d}^{4}{e}^{3}{x}^{3}-90090\,{a}^{2}bc{e}^{7}{x}^{2}+72072\,{a}^{2}{c}^{2}d{e}^{6}{x}^{2}-30030\,a{b}^{3}{e}^{7}{x}^{2}+144144\,a{b}^{2}cd{e}^{6}{x}^{2}-205920\,ab{c}^{2}{d}^{2}{e}^{5}{x}^{2}+91520\,a{c}^{3}{d}^{3}{e}^{4}{x}^{2}+12012\,{b}^{4}d{e}^{6}{x}^{2}-68640\,{b}^{3}c{d}^{2}{e}^{5}{x}^{2}+137280\,{b}^{2}{c}^{2}{d}^{3}{e}^{4}{x}^{2}-116480\,b{c}^{3}{d}^{4}{e}^{3}{x}^{2}+35840\,{c}^{4}{d}^{5}{e}^{2}{x}^{2}-60060\,{a}^{3}c{e}^{7}x-90090\,{a}^{2}{b}^{2}{e}^{7}x+360360\,{a}^{2}bcd{e}^{6}x-288288\,{a}^{2}{c}^{2}{d}^{2}{e}^{5}x+120120\,a{b}^{3}d{e}^{6}x-576576\,a{b}^{2}c{d}^{2}{e}^{5}x+823680\,ab{c}^{2}{d}^{3}{e}^{4}x-366080\,a{c}^{3}{d}^{4}{e}^{3}x-48048\,{b}^{4}{d}^{2}{e}^{5}x+274560\,{b}^{3}c{d}^{3}{e}^{4}x-549120\,{b}^{2}{c}^{2}{d}^{4}{e}^{3}x+465920\,b{c}^{3}{d}^{5}{e}^{2}x-143360\,{c}^{4}{d}^{6}ex+30030\,b{a}^{3}{e}^{7}-120120\,{a}^{3}cd{e}^{6}-180180\,{a}^{2}{b}^{2}d{e}^{6}+720720\,{a}^{2}bc{d}^{2}{e}^{5}-576576\,{a}^{2}{c}^{2}{d}^{3}{e}^{4}+240240\,a{b}^{3}{d}^{2}{e}^{5}-1153152\,a{b}^{2}c{d}^{3}{e}^{4}+1647360\,ab{c}^{2}{d}^{4}{e}^{3}-732160\,a{c}^{3}{d}^{5}{e}^{2}-96096\,{b}^{4}{d}^{3}{e}^{4}+549120\,{b}^{3}c{d}^{4}{e}^{3}-1098240\,{b}^{2}{c}^{2}{d}^{5}{e}^{2}+931840\,b{c}^{3}{d}^{6}e-286720\,{c}^{4}{d}^{7}}{15015\,{e}^{8}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15015/(e*x+d)^(1/2)*(-2310*c^4*e^7*x^7-9555*b*c^3*e^7*x^6+2940*c^4*d*e^6*x^6-10010*a*c^3*e^7*x^5-15015*b^2*
c^2*e^7*x^5+12740*b*c^3*d*e^6*x^5-3920*c^4*d^2*e^5*x^5-32175*a*b*c^2*e^7*x^4+14300*a*c^3*d*e^6*x^4-10725*b^3*c
*e^7*x^4+21450*b^2*c^2*d*e^6*x^4-18200*b*c^3*d^2*e^5*x^4+5600*c^4*d^3*e^4*x^4-18018*a^2*c^2*e^7*x^3-36036*a*b^
2*c*e^7*x^3+51480*a*b*c^2*d*e^6*x^3-22880*a*c^3*d^2*e^5*x^3-3003*b^4*e^7*x^3+17160*b^3*c*d*e^6*x^3-34320*b^2*c
^2*d^2*e^5*x^3+29120*b*c^3*d^3*e^4*x^3-8960*c^4*d^4*e^3*x^3-45045*a^2*b*c*e^7*x^2+36036*a^2*c^2*d*e^6*x^2-1501
5*a*b^3*e^7*x^2+72072*a*b^2*c*d*e^6*x^2-102960*a*b*c^2*d^2*e^5*x^2+45760*a*c^3*d^3*e^4*x^2+6006*b^4*d*e^6*x^2-
34320*b^3*c*d^2*e^5*x^2+68640*b^2*c^2*d^3*e^4*x^2-58240*b*c^3*d^4*e^3*x^2+17920*c^4*d^5*e^2*x^2-30030*a^3*c*e^
7*x-45045*a^2*b^2*e^7*x+180180*a^2*b*c*d*e^6*x-144144*a^2*c^2*d^2*e^5*x+60060*a*b^3*d*e^6*x-288288*a*b^2*c*d^2
*e^5*x+411840*a*b*c^2*d^3*e^4*x-183040*a*c^3*d^4*e^3*x-24024*b^4*d^2*e^5*x+137280*b^3*c*d^3*e^4*x-274560*b^2*c
^2*d^4*e^3*x+232960*b*c^3*d^5*e^2*x-71680*c^4*d^6*e*x+15015*a^3*b*e^7-60060*a^3*c*d*e^6-90090*a^2*b^2*d*e^6+36
0360*a^2*b*c*d^2*e^5-288288*a^2*c^2*d^3*e^4+120120*a*b^3*d^2*e^5-576576*a*b^2*c*d^3*e^4+823680*a*b*c^2*d^4*e^3
-366080*a*c^3*d^5*e^2-48048*b^4*d^3*e^4+274560*b^3*c*d^4*e^3-549120*b^2*c^2*d^5*e^2+465920*b*c^3*d^6*e-143360*
c^4*d^7)/e^8

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Maxima [A]  time = 1.02375, size = 882, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*((2310*(e*x + d)^(13/2)*c^4 - 9555*(2*c^4*d - b*c^3*e)*(e*x + d)^(11/2) + 5005*(14*c^4*d^2 - 14*b*c^3*
d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(9/2) - 10725*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 + 2*a*c^3
)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(7/2) + 3003*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c^2 + 2*a*
c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(5/2) - 15015*(14*
c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4 + 12*a*b^2
*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(3/2) + 15015*(14*c^4*d^6 - 42*b*c^3*d^5*e + 15*(3*
b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(a*
b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*sqrt(e*x + d))/e^7 + 15015*(2*c^4*d^7 - 7*b*c^3*d^6*e - a^
3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3
*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)/(sqrt(e*x + d)*e^7))/e

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Fricas [A]  time = 1.40168, size = 1527, normalized size = 3.63 \begin{align*} \frac{2 \,{\left (2310 \, c^{4} e^{7} x^{7} + 143360 \, c^{4} d^{7} - 465920 \, b c^{3} d^{6} e - 15015 \, a^{3} b e^{7} + 183040 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 274560 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 48048 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 120120 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + 30030 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 735 \,{\left (4 \, c^{4} d e^{6} - 13 \, b c^{3} e^{7}\right )} x^{6} + 35 \,{\left (112 \, c^{4} d^{2} e^{5} - 364 \, b c^{3} d e^{6} + 143 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 25 \,{\left (224 \, c^{4} d^{3} e^{4} - 728 \, b c^{3} d^{2} e^{5} + 286 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 429 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} +{\left (8960 \, c^{4} d^{4} e^{3} - 29120 \, b c^{3} d^{3} e^{4} + 11440 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 17160 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 3003 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} -{\left (17920 \, c^{4} d^{5} e^{2} - 58240 \, b c^{3} d^{4} e^{3} + 22880 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 34320 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 6006 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 15015 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} +{\left (71680 \, c^{4} d^{6} e - 232960 \, b c^{3} d^{5} e^{2} + 91520 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 137280 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 24024 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 60060 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 15015 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt{e x + d}}{15015 \,{\left (e^{9} x + d e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(2310*c^4*e^7*x^7 + 143360*c^4*d^7 - 465920*b*c^3*d^6*e - 15015*a^3*b*e^7 + 183040*(3*b^2*c^2 + 2*a*c^
3)*d^5*e^2 - 274560*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 48048*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 120120*(a*b^3
+ 3*a^2*b*c)*d^2*e^5 + 30030*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 735*(4*c^4*d*e^6 - 13*b*c^3*e^7)*x^6 + 35*(112*c^4
*d^2*e^5 - 364*b*c^3*d*e^6 + 143*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 25*(224*c^4*d^3*e^4 - 728*b*c^3*d^2*e^5 + 28
6*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 429*(b^3*c + 3*a*b*c^2)*e^7)*x^4 + (8960*c^4*d^4*e^3 - 29120*b*c^3*d^3*e^4 + 1
1440*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 17160*(b^3*c + 3*a*b*c^2)*d*e^6 + 3003*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7
)*x^3 - (17920*c^4*d^5*e^2 - 58240*b*c^3*d^4*e^3 + 22880*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 34320*(b^3*c + 3*a*b*
c^2)*d^2*e^5 + 6006*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 15015*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 + (71680*c^4*d^6
*e - 232960*b*c^3*d^5*e^2 + 91520*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 137280*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 24024*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 60060*(a*b^3 + 3*a^2*b*c)*d*e^6 + 15015*(3*a^2*b^2 + 2*a^3*c)*e^7)*x)*
sqrt(e*x + d)/(e^9*x + d*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.35476, size = 1350, normalized size = 3.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15015*(2310*(x*e + d)^(13/2)*c^4*e^96 - 19110*(x*e + d)^(11/2)*c^4*d*e^96 + 70070*(x*e + d)^(9/2)*c^4*d^2*e^
96 - 150150*(x*e + d)^(7/2)*c^4*d^3*e^96 + 210210*(x*e + d)^(5/2)*c^4*d^4*e^96 - 210210*(x*e + d)^(3/2)*c^4*d^
5*e^96 + 210210*sqrt(x*e + d)*c^4*d^6*e^96 + 9555*(x*e + d)^(11/2)*b*c^3*e^97 - 70070*(x*e + d)^(9/2)*b*c^3*d*
e^97 + 225225*(x*e + d)^(7/2)*b*c^3*d^2*e^97 - 420420*(x*e + d)^(5/2)*b*c^3*d^3*e^97 + 525525*(x*e + d)^(3/2)*
b*c^3*d^4*e^97 - 630630*sqrt(x*e + d)*b*c^3*d^5*e^97 + 15015*(x*e + d)^(9/2)*b^2*c^2*e^98 + 10010*(x*e + d)^(9
/2)*a*c^3*e^98 - 96525*(x*e + d)^(7/2)*b^2*c^2*d*e^98 - 64350*(x*e + d)^(7/2)*a*c^3*d*e^98 + 270270*(x*e + d)^
(5/2)*b^2*c^2*d^2*e^98 + 180180*(x*e + d)^(5/2)*a*c^3*d^2*e^98 - 450450*(x*e + d)^(3/2)*b^2*c^2*d^3*e^98 - 300
300*(x*e + d)^(3/2)*a*c^3*d^3*e^98 + 675675*sqrt(x*e + d)*b^2*c^2*d^4*e^98 + 450450*sqrt(x*e + d)*a*c^3*d^4*e^
98 + 10725*(x*e + d)^(7/2)*b^3*c*e^99 + 32175*(x*e + d)^(7/2)*a*b*c^2*e^99 - 60060*(x*e + d)^(5/2)*b^3*c*d*e^9
9 - 180180*(x*e + d)^(5/2)*a*b*c^2*d*e^99 + 150150*(x*e + d)^(3/2)*b^3*c*d^2*e^99 + 450450*(x*e + d)^(3/2)*a*b
*c^2*d^2*e^99 - 300300*sqrt(x*e + d)*b^3*c*d^3*e^99 - 900900*sqrt(x*e + d)*a*b*c^2*d^3*e^99 + 3003*(x*e + d)^(
5/2)*b^4*e^100 + 36036*(x*e + d)^(5/2)*a*b^2*c*e^100 + 18018*(x*e + d)^(5/2)*a^2*c^2*e^100 - 15015*(x*e + d)^(
3/2)*b^4*d*e^100 - 180180*(x*e + d)^(3/2)*a*b^2*c*d*e^100 - 90090*(x*e + d)^(3/2)*a^2*c^2*d*e^100 + 45045*sqrt
(x*e + d)*b^4*d^2*e^100 + 540540*sqrt(x*e + d)*a*b^2*c*d^2*e^100 + 270270*sqrt(x*e + d)*a^2*c^2*d^2*e^100 + 15
015*(x*e + d)^(3/2)*a*b^3*e^101 + 45045*(x*e + d)^(3/2)*a^2*b*c*e^101 - 90090*sqrt(x*e + d)*a*b^3*d*e^101 - 27
0270*sqrt(x*e + d)*a^2*b*c*d*e^101 + 45045*sqrt(x*e + d)*a^2*b^2*e^102 + 30030*sqrt(x*e + d)*a^3*c*e^102)*e^(-
104) + 2*(2*c^4*d^7 - 7*b*c^3*d^6*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e
^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*
e^6 + 2*a^3*c*d*e^6 - a^3*b*e^7)*e^(-8)/sqrt(x*e + d)