3.1222 \(\int (A+B x) (d+e x)^{5/2} (b x+c x^2)^2 \, dx\)
Optimal. Leaf size=267 \[ -\frac{2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac{2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac{2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac{2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac{2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac{2 B c^2 (d+e x)^{17/2}}{17 e^6} \]
[Out]
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(9/2))/(9*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(11/2))/(11*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))
*(d + e*x)^(13/2))/(13*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(15/2))/(15*e^6) + (2*B*c^2*(d + e*x)
^(17/2))/(17*e^6)
________________________________________________________________________________________
Rubi [A] time = 0.152087, antiderivative size = 267, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used =
{771} \[ -\frac{2 (d+e x)^{13/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{13 e^6}+\frac{2 (d+e x)^{11/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{11 e^6}-\frac{2 d^2 (d+e x)^{7/2} (B d-A e) (c d-b e)^2}{7 e^6}-\frac{2 c (d+e x)^{15/2} (-A c e-2 b B e+5 B c d)}{15 e^6}+\frac{2 d (d+e x)^{9/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{9 e^6}+\frac{2 B c^2 (d+e x)^{17/2}}{17 e^6} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(7/2))/(7*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(9/2))/(9*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(11/2))/(11*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))
*(d + e*x)^(13/2))/(13*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(15/2))/(15*e^6) + (2*B*c^2*(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 (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac{d^2 (B d-A e) (c d-b e)^2 (d+e x)^{5/2}}{e^5}+\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{7/2}}{e^5}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{9/2}}{e^5}+\frac{\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{11/2}}{e^5}+\frac{c (-5 B c d+2 b B e+A c e) (d+e x)^{13/2}}{e^5}+\frac{B c^2 (d+e x)^{15/2}}{e^5}\right ) \, dx\\ &=-\frac{2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{7/2}}{7 e^6}+\frac{2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{9/2}}{9 e^6}+\frac{2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^6}-\frac{2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{13/2}}{13 e^6}-\frac{2 c (5 B c d-2 b B e-A c e) (d+e x)^{15/2}}{15 e^6}+\frac{2 B c^2 (d+e x)^{17/2}}{17 e^6}\\ \end{align*}
Mathematica [A] time = 0.213362, size = 273, normalized size = 1.02 \[ \frac{2 (d+e x)^{7/2} \left (17 A e \left (65 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+30 b c e \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+c^2 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )+B \left (255 b^2 e^2 \left (56 d^2 e x-16 d^3-126 d e^2 x^2+231 e^3 x^3\right )+34 b c e \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )-5 c^2 \left (2016 d^3 e^2 x^2-3696 d^2 e^3 x^3-896 d^4 e x+256 d^5+6006 d e^4 x^4-9009 e^5 x^5\right )\right )\right )}{765765 e^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)*(d + e*x)^(5/2)*(b*x + c*x^2)^2,x]
[Out]
(2*(d + e*x)^(7/2)*(17*A*e*(65*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 30*b*c*e*(-16*d^3 + 56*d^2*e*x - 126*
d*e^2*x^2 + 231*e^3*x^3) + c^2*(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)) + B
*(255*b^2*e^2*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 34*b*c*e*(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*c^2*(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))))/(765765*e^6)
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Maple [A] time = 0.008, size = 341, normalized size = 1.3 \begin{align*}{\frac{90090\,B{c}^{2}{x}^{5}{e}^{5}+102102\,A{c}^{2}{e}^{5}{x}^{4}+204204\,Bbc{e}^{5}{x}^{4}-60060\,B{c}^{2}d{e}^{4}{x}^{4}+235620\,Abc{e}^{5}{x}^{3}-62832\,A{c}^{2}d{e}^{4}{x}^{3}+117810\,B{b}^{2}{e}^{5}{x}^{3}-125664\,Bbcd{e}^{4}{x}^{3}+36960\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+139230\,A{b}^{2}{e}^{5}{x}^{2}-128520\,Abcd{e}^{4}{x}^{2}+34272\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-64260\,B{b}^{2}d{e}^{4}{x}^{2}+68544\,Bbc{d}^{2}{e}^{3}{x}^{2}-20160\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-61880\,A{b}^{2}d{e}^{4}x+57120\,Abc{d}^{2}{e}^{3}x-15232\,A{c}^{2}{d}^{3}{e}^{2}x+28560\,B{b}^{2}{d}^{2}{e}^{3}x-30464\,Bbc{d}^{3}{e}^{2}x+8960\,B{c}^{2}{d}^{4}ex+17680\,A{b}^{2}{d}^{2}{e}^{3}-16320\,Abc{d}^{3}{e}^{2}+4352\,A{c}^{2}{d}^{4}e-8160\,B{b}^{2}{d}^{3}{e}^{2}+8704\,Bbc{d}^{4}e-2560\,B{c}^{2}{d}^{5}}{765765\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x)
[Out]
2/765765*(e*x+d)^(7/2)*(45045*B*c^2*e^5*x^5+51051*A*c^2*e^5*x^4+102102*B*b*c*e^5*x^4-30030*B*c^2*d*e^4*x^4+117
810*A*b*c*e^5*x^3-31416*A*c^2*d*e^4*x^3+58905*B*b^2*e^5*x^3-62832*B*b*c*d*e^4*x^3+18480*B*c^2*d^2*e^3*x^3+6961
5*A*b^2*e^5*x^2-64260*A*b*c*d*e^4*x^2+17136*A*c^2*d^2*e^3*x^2-32130*B*b^2*d*e^4*x^2+34272*B*b*c*d^2*e^3*x^2-10
080*B*c^2*d^3*e^2*x^2-30940*A*b^2*d*e^4*x+28560*A*b*c*d^2*e^3*x-7616*A*c^2*d^3*e^2*x+14280*B*b^2*d^2*e^3*x-152
32*B*b*c*d^3*e^2*x+4480*B*c^2*d^4*e*x+8840*A*b^2*d^2*e^3-8160*A*b*c*d^3*e^2+2176*A*c^2*d^4*e-4080*B*b^2*d^3*e^
2+4352*B*b*c*d^4*e-1280*B*c^2*d^5)/e^6
________________________________________________________________________________________
Maxima [A] time = 1.0962, size = 393, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (45045 \,{\left (e x + d\right )}^{\frac{17}{2}} B c^{2} - 51051 \,{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 58905 \,{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 69615 \,{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 109395 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{765765 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
2/765765*(45045*(e*x + d)^(17/2)*B*c^2 - 51051*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(15/2) + 58905*(10*
B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(13/2) - 69615*(10*B*c^2*d^3 - A*b^2*e^
3 - 6*(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(11/2) + 85085*(5*B*c^2*d^4 - 2*A*b^2*d*e
^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(9/2) - 109395*(B*c^2*d^5 - A*b^2*d^2*
e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(7/2))/e^6
________________________________________________________________________________________
Fricas [B] time = 1.78289, size = 1146, normalized size = 4.29 \begin{align*} \frac{2 \,{\left (45045 \, B c^{2} e^{8} x^{8} - 1280 \, B c^{2} d^{8} + 8840 \, A b^{2} d^{5} e^{3} + 2176 \,{\left (2 \, B b c + A c^{2}\right )} d^{7} e - 4080 \,{\left (B b^{2} + 2 \, A b c\right )} d^{6} e^{2} + 3003 \,{\left (35 \, B c^{2} d e^{7} + 17 \,{\left (2 \, B b c + A c^{2}\right )} e^{8}\right )} x^{7} + 231 \,{\left (275 \, B c^{2} d^{2} e^{6} + 527 \,{\left (2 \, B b c + A c^{2}\right )} d e^{7} + 255 \,{\left (B b^{2} + 2 \, A b c\right )} e^{8}\right )} x^{6} + 63 \,{\left (5 \, B c^{2} d^{3} e^{5} + 1105 \, A b^{2} e^{8} + 1207 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{6} + 2295 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{7}\right )} x^{5} - 35 \,{\left (10 \, B c^{2} d^{4} e^{4} - 5083 \, A b^{2} d e^{7} - 17 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{5} - 2703 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{6}\right )} x^{4} + 5 \,{\left (80 \, B c^{2} d^{5} e^{3} + 24973 \, A b^{2} d^{2} e^{6} - 136 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e^{4} + 255 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{5}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{6} e^{2} - 1105 \, A b^{2} d^{3} e^{5} - 272 \,{\left (2 \, B b c + A c^{2}\right )} d^{5} e^{3} + 510 \,{\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{4}\right )} x^{2} + 4 \,{\left (160 \, B c^{2} d^{7} e - 1105 \, A b^{2} d^{4} e^{4} - 272 \,{\left (2 \, B b c + A c^{2}\right )} d^{6} e^{2} + 510 \,{\left (B b^{2} + 2 \, A b c\right )} d^{5} e^{3}\right )} x\right )} \sqrt{e x + d}}{765765 \, e^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*(e*x+d)^(5/2)*(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
2/765765*(45045*B*c^2*e^8*x^8 - 1280*B*c^2*d^8 + 8840*A*b^2*d^5*e^3 + 2176*(2*B*b*c + A*c^2)*d^7*e - 4080*(B*b
^2 + 2*A*b*c)*d^6*e^2 + 3003*(35*B*c^2*d*e^7 + 17*(2*B*b*c + A*c^2)*e^8)*x^7 + 231*(275*B*c^2*d^2*e^6 + 527*(2
*B*b*c + A*c^2)*d*e^7 + 255*(B*b^2 + 2*A*b*c)*e^8)*x^6 + 63*(5*B*c^2*d^3*e^5 + 1105*A*b^2*e^8 + 1207*(2*B*b*c
+ A*c^2)*d^2*e^6 + 2295*(B*b^2 + 2*A*b*c)*d*e^7)*x^5 - 35*(10*B*c^2*d^4*e^4 - 5083*A*b^2*d*e^7 - 17*(2*B*b*c +
A*c^2)*d^3*e^5 - 2703*(B*b^2 + 2*A*b*c)*d^2*e^6)*x^4 + 5*(80*B*c^2*d^5*e^3 + 24973*A*b^2*d^2*e^6 - 136*(2*B*b
*c + A*c^2)*d^4*e^4 + 255*(B*b^2 + 2*A*b*c)*d^3*e^5)*x^3 - 3*(160*B*c^2*d^6*e^2 - 1105*A*b^2*d^3*e^5 - 272*(2*
B*b*c + A*c^2)*d^5*e^3 + 510*(B*b^2 + 2*A*b*c)*d^4*e^4)*x^2 + 4*(160*B*c^2*d^7*e - 1105*A*b^2*d^4*e^4 - 272*(2
*B*b*c + A*c^2)*d^6*e^2 + 510*(B*b^2 + 2*A*b*c)*d^5*e^3)*x)*sqrt(e*x + d)/e^6
________________________________________________________________________________________
Sympy [B] time = 54.7421, size = 1556, normalized size = 5.83 \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)**(5/2)*(c*x**2+b*x)**2,x)
[Out]
2*A*b**2*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*A*b**2*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*A*b**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**3 + 4*A*b*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 + 8*A*b*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 + 4*A*b*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 + 2*A*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 + 4*A*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 + 2*A*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 + 2*B*b**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 + 4*B*b**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 + 2*B*b**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 + 4*B*b*c*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 + 8*B*b*c*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 + 4*B*b*c*(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 + 2*B*c**2*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 + 4*B*c**2*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 + 2*B*c**2*(-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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Giac [B] time = 1.32659, size = 1866, normalized size = 6.99 \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)^(5/2)*(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
2/765765*(7293*(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^2*e^(-2) + 2431*(3
5*(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*b^2*d^2*e^(-3
) + 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*
c*d^2*e^(-3) + 442*(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*b*c*d^2*e^(-4) + 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)*A*c^2*d^2*e^(-4) + 85*(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^2*e^(-5) + 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*d*e^(-2) + 442*(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*b^2*d*e^(-3) + 8
84*(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*d*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)*B*b*c*d*e^(-4) + 1
70*(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 + 9
009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*A*c^2*d*e^(-4) + 34*(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*d*e^(-5) + 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)*A*b^2*e^(-2) + 85*(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*b^2*e^(-3) + 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)*
A*b*c*e^(-3) + 34*(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*b*c*e^(
-4) + 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)*A*c^2*e^(-4) + 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)*B*c^2*e^(-5))*e^(-1)