### 3.2283 $$\int (d+e x)^{3/2} (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=286 $\frac{6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}$

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(
7/2))/(7*e^7) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7)
- (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2
+ b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13/2))/(13*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) +
(2*c^3*(d + e*x)^(17/2))/(17*e^7)

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Rubi [A]  time = 0.129118, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {698} $\frac{6 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^7}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^7}+\frac{2 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^7) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(
7/2))/(7*e^7) + (2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(3*e^7)
- (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^7) + (6*c*(5*c^2*d^2
+ b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13/2))/(13*e^7) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(5*e^7) +
(2*c^3*(d + e*x)^(17/2))/(17*e^7)

Rule 698

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

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{e^6}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^6}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{7/2}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{9/2}}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{11/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^6}+\frac{c^3 (d+e x)^{15/2}}{e^6}\right ) \, dx\\ &=\frac{2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^7}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^7}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{3 e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^7}+\frac{6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{13/2}}{13 e^7}-\frac{2 c^2 (2 c d-b e) (d+e x)^{15/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{17/2}}{17 e^7}\\ \end{align*}

Mathematica [A]  time = 0.882017, size = 320, normalized size = 1.12 $\frac{2 \left ((d+e x)^{5/2} (a+x (b+c x))^3-\frac{2 (d+e x)^{7/2} \left (-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (-56 d^2 e x+16 d^3+126 d e^2 x^2-231 e^3 x^3\right )\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+17 c^2 e \left (12 a e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+b \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-2 c^3 \left (2016 d^3 e^2 x^2-3696 d^2 e^3 x^3-896 d^4 e x+256 d^5+6006 d e^4 x^4-9009 e^5 x^5\right )\right )}{51051 e^6}\right )}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*((d + e*x)^(5/2)*(a + x*(b + c*x))^3 - (2*(d + e*x)^(7/2)*(-2*c^3*(256*d^5 - 896*d^4*e*x + 2016*d^3*e^2*x^2
- 3696*d^2*e^3*x^3 + 6006*d*e^4*x^4 - 9009*e^5*x^5) + 221*b*e^3*(99*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(
8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 34*c*e^2*(143*a^2*e^2*(2*d - 7*e*x) - 39*a*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^
2) + 6*b^2*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3)) + 17*c^2*e*(12*a*e*(-16*d^3 + 56*d^2*e*x - 126
*d*e^2*x^2 + 231*e^3*x^3) + b*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(5
1051*e^6)))/(5*e)

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Maple [A]  time = 0.045, size = 495, normalized size = 1.7 \begin{align*}{\frac{30030\,{c}^{3}{x}^{6}{e}^{6}+102102\,b{c}^{2}{e}^{6}{x}^{5}-24024\,{c}^{3}d{e}^{5}{x}^{5}+117810\,a{c}^{2}{e}^{6}{x}^{4}+117810\,{b}^{2}c{e}^{6}{x}^{4}-78540\,b{c}^{2}d{e}^{5}{x}^{4}+18480\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}+278460\,abc{e}^{6}{x}^{3}-85680\,a{c}^{2}d{e}^{5}{x}^{3}+46410\,{b}^{3}{e}^{6}{x}^{3}-85680\,{b}^{2}cd{e}^{5}{x}^{3}+57120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}-13440\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+170170\,{a}^{2}c{e}^{6}{x}^{2}+170170\,a{b}^{2}{e}^{6}{x}^{2}-185640\,abcd{e}^{5}{x}^{2}+57120\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-30940\,{b}^{3}d{e}^{5}{x}^{2}+57120\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-38080\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+8960\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+218790\,{a}^{2}b{e}^{6}x-97240\,{a}^{2}cd{e}^{5}x-97240\,a{b}^{2}d{e}^{5}x+106080\,abc{d}^{2}{e}^{4}x-32640\,a{c}^{2}{d}^{3}{e}^{3}x+17680\,{b}^{3}{d}^{2}{e}^{4}x-32640\,{b}^{2}c{d}^{3}{e}^{3}x+21760\,b{c}^{2}{d}^{4}{e}^{2}x-5120\,{c}^{3}{d}^{5}ex+102102\,{a}^{3}{e}^{6}-87516\,{a}^{2}bd{e}^{5}+38896\,{a}^{2}c{d}^{2}{e}^{4}+38896\,a{b}^{2}{d}^{2}{e}^{4}-42432\,abc{d}^{3}{e}^{3}+13056\,a{c}^{2}{d}^{4}{e}^{2}-7072\,{b}^{3}{d}^{3}{e}^{3}+13056\,{b}^{2}c{d}^{4}{e}^{2}-8704\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(e*x+d)^(5/2)*(15015*c^3*e^6*x^6+51051*b*c^2*e^6*x^5-12012*c^3*d*e^5*x^5+58905*a*c^2*e^6*x^4+58905*b^
2*c*e^6*x^4-39270*b*c^2*d*e^5*x^4+9240*c^3*d^2*e^4*x^4+139230*a*b*c*e^6*x^3-42840*a*c^2*d*e^5*x^3+23205*b^3*e^
6*x^3-42840*b^2*c*d*e^5*x^3+28560*b*c^2*d^2*e^4*x^3-6720*c^3*d^3*e^3*x^3+85085*a^2*c*e^6*x^2+85085*a*b^2*e^6*x
^2-92820*a*b*c*d*e^5*x^2+28560*a*c^2*d^2*e^4*x^2-15470*b^3*d*e^5*x^2+28560*b^2*c*d^2*e^4*x^2-19040*b*c^2*d^3*e
^3*x^2+4480*c^3*d^4*e^2*x^2+109395*a^2*b*e^6*x-48620*a^2*c*d*e^5*x-48620*a*b^2*d*e^5*x+53040*a*b*c*d^2*e^4*x-1
6320*a*c^2*d^3*e^3*x+8840*b^3*d^2*e^4*x-16320*b^2*c*d^3*e^3*x+10880*b*c^2*d^4*e^2*x-2560*c^3*d^5*e*x+51051*a^3
*e^6-43758*a^2*b*d*e^5+19448*a^2*c*d^2*e^4+19448*a*b^2*d^2*e^4-21216*a*b*c*d^3*e^3+6528*a*c^2*d^4*e^2-3536*b^3
*d^3*e^3+6528*b^2*c*d^4*e^2-4352*b*c^2*d^5*e+1024*c^3*d^6)/e^7

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Maxima [A]  time = 1.03337, size = 549, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} - 51051 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 23205 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2) + 58905*(5*c^3*d^2 - 5*b*c^2
*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(13/2) - 23205*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 -
(b^3 + 6*a*b*c)*e^3)*(e*x + d)^(11/2) + 85085*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3
+ 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(9/2) - 109395*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b
^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(7/2) + 51051*(c^3*d^6 -
3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(a*b^2 + a^2
*c)*d^2*e^4)*(e*x + d)^(5/2))/e^7

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Fricas [B]  time = 2.04237, size = 1426, normalized size = 4.99 \begin{align*} \frac{2 \,{\left (15015 \, c^{3} e^{8} x^{8} + 1024 \, c^{3} d^{8} - 4352 \, b c^{2} d^{7} e - 43758 \, a^{2} b d^{3} e^{5} + 51051 \, a^{3} d^{2} e^{6} + 6528 \,{\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} - 3536 \,{\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} + 19448 \,{\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \,{\left (6 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (c^{3} d^{2} e^{6} + 272 \, b c^{2} d e^{7} + 255 \,{\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} - 21 \,{\left (12 \, c^{3} d^{3} e^{5} - 51 \, b c^{2} d^{2} e^{6} - 3570 \,{\left (b^{2} c + a c^{2}\right )} d e^{7} - 1105 \,{\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} + 35 \,{\left (8 \, c^{3} d^{4} e^{4} - 34 \, b c^{2} d^{3} e^{5} + 51 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} + 884 \,{\left (b^{3} + 6 \, a b c\right )} d e^{7} + 2431 \,{\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{5} e^{3} - 272 \, b c^{2} d^{4} e^{4} - 21879 \, a^{2} b e^{8} + 408 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} - 221 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} - 24310 \,{\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} + 3 \,{\left (128 \, c^{3} d^{6} e^{2} - 544 \, b c^{2} d^{5} e^{3} + 58344 \, a^{2} b d e^{7} + 17017 \, a^{3} e^{8} + 816 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 442 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} + 2431 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} -{\left (512 \, c^{3} d^{7} e - 2176 \, b c^{2} d^{6} e^{2} - 21879 \, a^{2} b d^{2} e^{6} - 102102 \, a^{3} d e^{7} + 3264 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 1768 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 9724 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*c^3*e^8*x^8 + 1024*c^3*d^8 - 4352*b*c^2*d^7*e - 43758*a^2*b*d^3*e^5 + 51051*a^3*d^2*e^6 + 6528
*(b^2*c + a*c^2)*d^6*e^2 - 3536*(b^3 + 6*a*b*c)*d^5*e^3 + 19448*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(6*c^3*d*e^7 +
17*b*c^2*e^8)*x^7 + 231*(c^3*d^2*e^6 + 272*b*c^2*d*e^7 + 255*(b^2*c + a*c^2)*e^8)*x^6 - 21*(12*c^3*d^3*e^5 - 5
1*b*c^2*d^2*e^6 - 3570*(b^2*c + a*c^2)*d*e^7 - 1105*(b^3 + 6*a*b*c)*e^8)*x^5 + 35*(8*c^3*d^4*e^4 - 34*b*c^2*d^
3*e^5 + 51*(b^2*c + a*c^2)*d^2*e^6 + 884*(b^3 + 6*a*b*c)*d*e^7 + 2431*(a*b^2 + a^2*c)*e^8)*x^4 - 5*(64*c^3*d^5
*e^3 - 272*b*c^2*d^4*e^4 - 21879*a^2*b*e^8 + 408*(b^2*c + a*c^2)*d^3*e^5 - 221*(b^3 + 6*a*b*c)*d^2*e^6 - 24310
*(a*b^2 + a^2*c)*d*e^7)*x^3 + 3*(128*c^3*d^6*e^2 - 544*b*c^2*d^5*e^3 + 58344*a^2*b*d*e^7 + 17017*a^3*e^8 + 816
*(b^2*c + a*c^2)*d^4*e^4 - 442*(b^3 + 6*a*b*c)*d^3*e^5 + 2431*(a*b^2 + a^2*c)*d^2*e^6)*x^2 - (512*c^3*d^7*e -
2176*b*c^2*d^6*e^2 - 21879*a^2*b*d^2*e^6 - 102102*a^3*d*e^7 + 3264*(b^2*c + a*c^2)*d^5*e^3 - 1768*(b^3 + 6*a*b
*c)*d^4*e^4 + 9724*(a*b^2 + a^2*c)*d^3*e^5)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 39.5039, size = 1411, normalized size = 4.93 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 6*a**2*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 6*a**2*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 + 6*a**2*c*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 + 6*a**2*c*(-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 + 6*a*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 + 6*a*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 + 12*a*b*c*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 + 12*a*b*c*(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 + 6*a*c**2*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)/1
1)/e**5 + 6*a*c**2*(-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 + 2*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 + 2*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 + 6*b**2*c*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 + 6*b**2*c*(-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 + 6
*b*c**2*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 + 6*b*c**2*(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 + 2*c**3*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)/11 - 6*d*(d + e*x
)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*c**3*(-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)/1
3 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7

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Giac [B]  time = 1.1873, size = 1690, 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((e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/765765*(153153*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b*d*e^(-1) + 21879*(15*(x*e + d)^(7/2) - 42*(x*
e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b^2*d*e^(-2) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 3
5*(x*e + d)^(3/2)*d^2)*a^2*c*d*e^(-2) + 2431*(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)*b^3*d*e^(-3) + 14586*(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*b*c*d*e^(-3) + 663*(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)*b^2*c*d*e^(-4) + 663*(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*c^2*d*e^(-4) + 255*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 1
2870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b*c^2*d*e^(-5) + 17*(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)*c^3*d*e^(-6) + 255255*(x*e + d)^(3/2)
*a^3*d + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*e^(-1) + 7293*(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*b^2*e^(-2) + 7293
*(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^2*c*e^(-2)
+ 221*(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)*b^3*e^(-3) + 1326*(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*b*c*e^(-3) + 255*(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)*b^2*c*e^(-4) + 255*(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*c^2*e^(-4) + 51*(30
03*(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 + 965
25*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b*c^2*e^(-5) + 7*(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)*c^3*e^(
-6) + 51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^3)*e^(-1)