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3.15.9 \(\int (b+2 c x) (d+e x)^{5/2} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=252 \[ \frac {8 c (d+e x)^{13/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{13 e^6}-\frac {2 (d+e x)^{11/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{11 e^6}+\frac {4 (d+e x)^{9/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6}-\frac {2 c^2 (d+e x)^{15/2} (2 c d-b e)}{3 e^6}+\frac {4 c^3 (d+e x)^{17/2}}{17 e^6} \]

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Rubi [A]  time = 0.17, antiderivative size = 252, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.41, size = 291, normalized size = 1.15 \begin {gather*} \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 (16 d^3-56 d^2 e x+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 (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b \left (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\right )\right )-2 c^3 \left (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\right )\right )}{153153 e^6} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.20, size = 425, normalized size = 1.69 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (21879 a^2 b e^5+34034 a^2 c e^4 (d+e x)-43758 a^2 c d e^4+34034 a b^2 e^4 (d+e x)-43758 a b^2 d e^4+131274 a b c d^2 e^3-204204 a b c d e^3 (d+e x)+83538 a b c e^3 (d+e x)^2-87516 a c^2 d^3 e^2+204204 a c^2 d^2 e^2 (d+e x)-167076 a c^2 d e^2 (d+e x)^2+47124 a c^2 e^2 (d+e x)^3+21879 b^3 d^2 e^3-34034 b^3 d e^3 (d+e x)+13923 b^3 e^3 (d+e x)^2-87516 b^2 c d^3 e^2+204204 b^2 c d^2 e^2 (d+e x)-167076 b^2 c d e^2 (d+e x)^2+47124 b^2 c e^2 (d+e x)^3+109395 b c^2 d^4 e-340340 b c^2 d^3 e (d+e x)+417690 b c^2 d^2 e (d+e x)^2-235620 b c^2 d e (d+e x)^3+51051 b c^2 e (d+e x)^4-43758 c^3 d^5+170170 c^3 d^4 (d+e x)-278460 c^3 d^3 (d+e x)^2+235620 c^3 d^2 (d+e x)^3-102102 c^3 d (d+e x)^4+18018 c^3 (d+e x)^5\right )}{153153 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(-43758*c^3*d^5 + 109395*b*c^2*d^4*e - 87516*b^2*c*d^3*e^2 - 87516*a*c^2*d^3*e^2 + 21879*b^
3*d^2*e^3 + 131274*a*b*c*d^2*e^3 - 43758*a*b^2*d*e^4 - 43758*a^2*c*d*e^4 + 21879*a^2*b*e^5 + 170170*c^3*d^4*(d
 + e*x) - 340340*b*c^2*d^3*e*(d + e*x) + 204204*b^2*c*d^2*e^2*(d + e*x) + 204204*a*c^2*d^2*e^2*(d + e*x) - 340
34*b^3*d*e^3*(d + e*x) - 204204*a*b*c*d*e^3*(d + e*x) + 34034*a*b^2*e^4*(d + e*x) + 34034*a^2*c*e^4*(d + e*x)
- 278460*c^3*d^3*(d + e*x)^2 + 417690*b*c^2*d^2*e*(d + e*x)^2 - 167076*b^2*c*d*e^2*(d + e*x)^2 - 167076*a*c^2*
d*e^2*(d + e*x)^2 + 13923*b^3*e^3*(d + e*x)^2 + 83538*a*b*c*e^3*(d + e*x)^2 + 235620*c^3*d^2*(d + e*x)^3 - 235
620*b*c^2*d*e*(d + e*x)^3 + 47124*b^2*c*e^2*(d + e*x)^3 + 47124*a*c^2*e^2*(d + e*x)^3 - 102102*c^3*d*(d + e*x)
^4 + 51051*b*c^2*e*(d + e*x)^4 + 18018*c^3*(d + e*x)^5))/(153153*e^6)

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fricas [B]  time = 0.41, size = 590, normalized size = 2.34 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.33, size = 2447, normalized size = 9.71

result too large to display

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*(510510*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b^2*d^3*e^(-1) + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e
 + d)*d)*a^2*c*d^3*e^(-1) + 51051*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^3*d^3*e^
(-2) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*c*d^3*e^(-2) + 87516*(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)*b^2*c*d^3*e^(-3) + 87516
*(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*c^2*d^3*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*s
qrt(x*e + d)*d^4)*b*c^2*d^3*e^(-4) + 2210*(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)*c^3*d^3*e^(-5) + 306306*(3*(
x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*d^2*e^(-1) + 306306*(3*(x*e + d)^(5/2) - 1
0*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d^2*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)*b^3*d^2*e^(-2) + 393822*(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*b*c*d^2*e^(-2) + 29172*(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)*b^2*c*d^2*e^(-3) +
29172*(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*sq
rt(x*e + d)*d^4)*a*c^2*d^2*e^(-3) + 16575*(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)*b*c^2*d^2*e^(-4) + 1530*(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)*c^3*d^2*e^(-5) + 765765*sqrt(x*e + d)*a^2*
b*d^3 + 765765*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b*d^2 + 131274*(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*b^2*d*e^(-1) + 131274*(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^2*c*d*e^(-1) + 7293*(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)*b^3*d*e^(-2) + 4375
8*(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*b*c*d*e^(-2) + 13260*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1
386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^2*c*d*e^(-3) + 13260*(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*c^2*d*e^(-3) + 3825*(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)*b*c^2*d*e^(-4) + 714*(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)*c^3*d*e^(-5) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d
)*d^2)*a^2*b*d + 4862*(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*b^2*e^(-1) + 4862*(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^2*c*e^(-1) + 1105*(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)*b^3*e^(-2) + 6630*(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*b*c*e^(-2) + 1020*(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)*b^2*c*e^(-3) + 1020*(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*c^2*e^(-3) + 595*(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 + 15
015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b*c^2*e^(-4) + 14*(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)*c^3*e^(-
5) + 21879*(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^2*b)*e
^(-1)

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maple [A]  time = 0.06, size = 359, normalized size = 1.42 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (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 -37128 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}\right )}{153153 e^{6}} \end {gather*}

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 = 0.60, size = 308, normalized size = 1.22 \begin {gather*} \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 {gather*}

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

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mupad [B]  time = 0.10, size = 267, normalized size = 1.06 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{9\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{13\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x)

[Out]

((d + e*x)^(9/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 -
 40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(9*e^6) + (4*c^3*(d + e*x)^(17/2))/(17*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d +
 e*x)^(15/2))/(15*e^6) + ((d + e*x)^(13/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(13*e^6) +
 (2*(b*e - 2*c*d)*(d + e*x)^(11/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(11*e^6) + (2*(b*e - 2*c*d
)*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(7*e^6)

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sympy [A]  time = 70.39, size = 1860, normalized size = 7.38

result too large to display

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

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