3.1601 \(\int (b+2 c x) (d+e x)^{5/2} (a+b x+c x^2)^2 \, dx\)
Optimal. Leaf size=252 \[ \frac{8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac{4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac{4 c^3 (d+e x)^{17/2}}{17 e^6} \]
[Out]
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^6) + (4*(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))/(9*e^6) - (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^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13/2))/(13*e
^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(3*e^6) + (4*c^3*(d + e*x)^(17/2))/(17*e^6)
________________________________________________________________________________________
Rubi [A] time = 0.172638, antiderivative size = 252, 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} \[ \frac{8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac{2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac{4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac{2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac{2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac{4 c^3 (d+e x)^{17/2}}{17 e^6} \]
Antiderivative was successfully verified.
[In]
Int[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
[Out]
(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^6) + (4*(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))/(9*e^6) - (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^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(13/2))/(13*e
^6) - (2*c^2*(2*c*d - b*e)*(d + e*x)^(15/2))/(3*e^6) + (4*c^3*(d + e*x)^(17/2))/(17*e^6)
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 (b+2 c x) (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^5}+\frac{2 \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^5}+\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^5}+\frac{4 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^5}-\frac{5 c^2 (2 c d-b e) (d+e x)^{13/2}}{e^5}+\frac{2 c^3 (d+e x)^{15/2}}{e^5}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^6}+\frac{4 \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}}{9 e^6}-\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^6}+\frac{8 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^6}-\frac{2 c^2 (2 c d-b e) (d+e x)^{15/2}}{3 e^6}+\frac{4 c^3 (d+e x)^{17/2}}{17 e^6}\\ \end{align*}
Mathematica [A] time = 0.42286, size = 291, normalized size = 1.15 \[ \frac{2 (d+e x)^{7/2} \left (-34 c e^2 \left (143 a^2 e^2 (2 d-7 e x)-39 a b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+6 b^2 \left (-56 d^2 e x+16 d^3+126 d e^2 x^2-231 e^3 x^3\right )\right )+221 b e^3 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+17 c^2 e \left (12 a e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+b \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )-2 c^3 \left (2016 d^3 e^2 x^2-3696 d^2 e^3 x^3-896 d^4 e x+256 d^5+6006 d e^4 x^4-9009 e^5 x^5\right )\right )}{153153 e^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
[Out]
(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 - 900
9*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 - 4
48*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4))))/(153153*e^6)
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Maple [A] time = 0.007, size = 359, normalized size = 1.4 \begin{align*}{\frac{36036\,{c}^{3}{x}^{5}{e}^{5}+102102\,b{c}^{2}{e}^{5}{x}^{4}-24024\,{c}^{3}d{e}^{4}{x}^{4}+94248\,a{c}^{2}{e}^{5}{x}^{3}+94248\,{b}^{2}c{e}^{5}{x}^{3}-62832\,b{c}^{2}d{e}^{4}{x}^{3}+14784\,{c}^{3}{d}^{2}{e}^{3}{x}^{3}+167076\,abc{e}^{5}{x}^{2}-51408\,a{c}^{2}d{e}^{4}{x}^{2}+27846\,{b}^{3}{e}^{5}{x}^{2}-51408\,{b}^{2}cd{e}^{4}{x}^{2}+34272\,b{c}^{2}{d}^{2}{e}^{3}{x}^{2}-8064\,{c}^{3}{d}^{3}{e}^{2}{x}^{2}+68068\,{a}^{2}c{e}^{5}x+68068\,a{b}^{2}{e}^{5}x-74256\,abcd{e}^{4}x+22848\,a{c}^{2}{d}^{2}{e}^{3}x-12376\,{b}^{3}d{e}^{4}x+22848\,{b}^{2}c{d}^{2}{e}^{3}x-15232\,b{c}^{2}{d}^{3}{e}^{2}x+3584\,{c}^{3}{d}^{4}ex+43758\,b{a}^{2}{e}^{5}-19448\,{a}^{2}cd{e}^{4}-19448\,a{b}^{2}d{e}^{4}+21216\,abc{d}^{2}{e}^{3}-6528\,a{c}^{2}{d}^{3}{e}^{2}+3536\,{b}^{3}{d}^{2}{e}^{3}-6528\,{b}^{2}c{d}^{3}{e}^{2}+4352\,b{c}^{2}{d}^{4}e-1024\,{c}^{3}{d}^{5}}{153153\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x)
[Out]
2/153153*(e*x+d)^(7/2)*(18018*c^3*e^5*x^5+51051*b*c^2*e^5*x^4-12012*c^3*d*e^4*x^4+47124*a*c^2*e^5*x^3+47124*b^
2*c*e^5*x^3-31416*b*c^2*d*e^4*x^3+7392*c^3*d^2*e^3*x^3+83538*a*b*c*e^5*x^2-25704*a*c^2*d*e^4*x^2+13923*b^3*e^5
*x^2-25704*b^2*c*d*e^4*x^2+17136*b*c^2*d^2*e^3*x^2-4032*c^3*d^3*e^2*x^2+34034*a^2*c*e^5*x+34034*a*b^2*e^5*x-37
128*a*b*c*d*e^4*x+11424*a*c^2*d^2*e^3*x-6188*b^3*d*e^4*x+11424*b^2*c*d^2*e^3*x-7616*b*c^2*d^3*e^2*x+1792*c^3*d
^4*e*x+21879*a^2*b*e^5-9724*a^2*c*d*e^4-9724*a*b^2*d*e^4+10608*a*b*c*d^2*e^3-3264*a*c^2*d^3*e^2+1768*b^3*d^2*e
^3-3264*b^2*c*d^3*e^2+2176*b*c^2*d^4*e-512*c^3*d^5)/e^6
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Maxima [A] time = 0.995289, size = 416, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (18018 \,{\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}} + 47124 \,{\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}} - 13923 \,{\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}} + 34034 \,{\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}} - 21879 \,{\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}}\right )}}{153153 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
2/153153*(18018*(e*x + d)^(17/2)*c^3 - 51051*(2*c^3*d - b*c^2*e)*(e*x + d)^(15/2) + 47124*(5*c^3*d^2 - 5*b*c^2
*d*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(13/2) - 13923*(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) + 34034*(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) - 21879*(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))/e^6
________________________________________________________________________________________
Fricas [B] time = 1.27699, size = 1362, normalized size = 5.4 \begin{align*} \frac{2 \,{\left (18018 \, c^{3} e^{8} x^{8} - 512 \, c^{3} d^{8} + 2176 \, b c^{2} d^{7} e + 21879 \, a^{2} b d^{3} e^{5} - 3264 \,{\left (b^{2} c + a c^{2}\right )} d^{6} e^{2} + 1768 \,{\left (b^{3} + 6 \, a b c\right )} d^{5} e^{3} - 9724 \,{\left (a b^{2} + a^{2} c\right )} d^{4} e^{4} + 3003 \,{\left (14 \, c^{3} d e^{7} + 17 \, b c^{2} e^{8}\right )} x^{7} + 231 \,{\left (110 \, c^{3} d^{2} e^{6} + 527 \, b c^{2} d e^{7} + 204 \,{\left (b^{2} c + a c^{2}\right )} e^{8}\right )} x^{6} + 63 \,{\left (2 \, c^{3} d^{3} e^{5} + 1207 \, b c^{2} d^{2} e^{6} + 1836 \,{\left (b^{2} c + a c^{2}\right )} d e^{7} + 221 \,{\left (b^{3} + 6 \, a b c\right )} e^{8}\right )} x^{5} - 7 \,{\left (20 \, c^{3} d^{4} e^{4} - 85 \, b c^{2} d^{3} e^{5} - 10812 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{6} - 5083 \,{\left (b^{3} + 6 \, a b c\right )} d e^{7} - 4862 \,{\left (a b^{2} + a^{2} c\right )} e^{8}\right )} x^{4} +{\left (160 \, c^{3} d^{5} e^{3} - 680 \, b c^{2} d^{4} e^{4} + 21879 \, a^{2} b e^{8} + 1020 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{5} + 24973 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{6} + 92378 \,{\left (a b^{2} + a^{2} c\right )} d e^{7}\right )} x^{3} - 3 \,{\left (64 \, c^{3} d^{6} e^{2} - 272 \, b c^{2} d^{5} e^{3} - 21879 \, a^{2} b d e^{7} + 408 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{4} - 221 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{5} - 24310 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{6}\right )} x^{2} +{\left (256 \, c^{3} d^{7} e - 1088 \, b c^{2} d^{6} e^{2} + 65637 \, a^{2} b d^{2} e^{6} + 1632 \,{\left (b^{2} c + a c^{2}\right )} d^{5} e^{3} - 884 \,{\left (b^{3} + 6 \, a b c\right )} d^{4} e^{4} + 4862 \,{\left (a b^{2} + a^{2} c\right )} d^{3} e^{5}\right )} x\right )} \sqrt{e x + d}}{153153 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
2/153153*(18018*c^3*e^8*x^8 - 512*c^3*d^8 + 2176*b*c^2*d^7*e + 21879*a^2*b*d^3*e^5 - 3264*(b^2*c + a*c^2)*d^6*
e^2 + 1768*(b^3 + 6*a*b*c)*d^5*e^3 - 9724*(a*b^2 + a^2*c)*d^4*e^4 + 3003*(14*c^3*d*e^7 + 17*b*c^2*e^8)*x^7 + 2
31*(110*c^3*d^2*e^6 + 527*b*c^2*d*e^7 + 204*(b^2*c + a*c^2)*e^8)*x^6 + 63*(2*c^3*d^3*e^5 + 1207*b*c^2*d^2*e^6
+ 1836*(b^2*c + a*c^2)*d*e^7 + 221*(b^3 + 6*a*b*c)*e^8)*x^5 - 7*(20*c^3*d^4*e^4 - 85*b*c^2*d^3*e^5 - 10812*(b^
2*c + a*c^2)*d^2*e^6 - 5083*(b^3 + 6*a*b*c)*d*e^7 - 4862*(a*b^2 + a^2*c)*e^8)*x^4 + (160*c^3*d^5*e^3 - 680*b*c
^2*d^4*e^4 + 21879*a^2*b*e^8 + 1020*(b^2*c + a*c^2)*d^3*e^5 + 24973*(b^3 + 6*a*b*c)*d^2*e^6 + 92378*(a*b^2 + a
^2*c)*d*e^7)*x^3 - 3*(64*c^3*d^6*e^2 - 272*b*c^2*d^5*e^3 - 21879*a^2*b*d*e^7 + 408*(b^2*c + a*c^2)*d^4*e^4 - 2
21*(b^3 + 6*a*b*c)*d^3*e^5 - 24310*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + (256*c^3*d^7*e - 1088*b*c^2*d^6*e^2 + 65637*
a^2*b*d^2*e^6 + 1632*(b^2*c + a*c^2)*d^5*e^3 - 884*(b^3 + 6*a*b*c)*d^4*e^4 + 4862*(a*b^2 + a^2*c)*d^3*e^5)*x)*
sqrt(e*x + d)/e^6
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Sympy [A] time = 53.7551, size = 1860, normalized size = 7.38 \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)*(e*x+d)**(5/2)*(c*x**2+b*x+a)**2,x)
[Out]
a**2*b*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**2*b*d*(-d*(d + e*x)**(3/
2)/3 + (d + e*x)**(5/2)/5)/e + 2*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 + 4*a**2*c*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 8*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**2 + 4*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**2 + 4*a*b**2*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(
5/2)/5)/e**2 + 8*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**2 + 4*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**2
+ 12*a*b*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 24*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**3 + 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**3 + 8*a*c**2*d**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**4 + 16*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)/11)/e**4 + 8*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**4 + 2*b**3*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)*
*(7/2)/7)/e**3 + 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**3 + 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**3 + 8*b**2*c*d**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**4 + 16*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**4 + 8
*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**4 + 10*b*c**2*d**2*(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 + 20*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**5 + 10*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**5 + 4*c**3*d**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**6 + 8*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**6 +
4*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)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(
17/2)/17)/e**6
________________________________________________________________________________________
Giac [B] time = 1.28345, size = 2233, normalized size = 8.86 \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)*(e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
2/765765*(102102*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b^2*d^2*e^(-1) + 102102*(3*(x*e + d)^(5/2) - 5*(x
*e + d)^(3/2)*d)*a^2*c*d^2*e^(-1) + 7293*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*
b^3*d^2*e^(-2) + 43758*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*c*d^2*e^(-2) +
9724*(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^2*c*d
^2*e^(-3) + 9724*(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*c^2*d^2*e^(-3) + 1105*(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*c^2*d^2*e^(-4) + 170*(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)*c^3*d^2*e^(-5) + 255255*(x*e + d)^(3/2)*a^2*b*d^2 + 29172*(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^(-1) + 29172*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/
2)*d^2)*a^2*c*d*e^(-1) + 4862*(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^(-2) + 29172*(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^(-2) + 1768*(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^(-3) + 1768*(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^(-3) + 850*(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*c^2*d*e^(-4) + 68*(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^(-5) + 102102*(3*(x*e + d)^(5/2) - 5*(x*e
+ d)^(3/2)*d)*a^2*b*d + 4862*(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^(-1) + 4862*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 10
5*(x*e + d)^(3/2)*d^3)*a^2*c*e^(-1) + 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^(-2) + 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^(-2)
+ 340*(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^(-3) + 340*(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^(-3) + 85*(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)*b*c^2*e^(-4) + 14*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 34807
5*(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^(-5) + 7293*(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)