3.1435 \(\int (A+B x) \sqrt{d+e x} (a+c x^2)^2 \, dx\)

Optimal. Leaf size=218 \[ \frac{4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}-\frac{4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e
^2)*(d + e*x)^(5/2))/(5*e^6) - (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(7/2))/(7*e^6)
+ (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(9*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(11/2))/(1
1*e^6) + (2*B*c^2*(d + e*x)^(13/2))/(13*e^6)

________________________________________________________________________________________

Rubi [A]  time = 0.127017, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {772} \[ \frac{4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}-\frac{4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (2*(c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e
^2)*(d + e*x)^(5/2))/(5*e^6) - (4*c*(5*B*c*d^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(7/2))/(7*e^6)
+ (4*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(9*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(11/2))/(1
1*e^6) + (2*B*c^2*(d + e*x)^(13/2))/(13*e^6)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.208126, size = 214, normalized size = 0.98 \[ \frac{2 (d+e x)^{3/2} \left (13 A e \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (3003 a^2 e^4 (3 e x-2 d)+286 a c e^2 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )-5 c^2 \left (480 d^3 e^2 x^2-560 d^2 e^3 x^3-384 d^4 e x+256 d^5+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(13*A*e*(1155*a^2*e^4 + 66*a*c*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 - 192*d^3
*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) + B*(3003*a^2*e^4*(-2*d + 3*e*x) + 286*a*c*e^2*(-16*d^3
+ 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) - 5*c^2*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3
+ 630*d*e^4*x^4 - 693*e^5*x^5))))/(45045*e^6)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 259, normalized size = 1.2 \begin{align*}{\frac{6930\,B{c}^{2}{x}^{5}{e}^{5}+8190\,A{c}^{2}{e}^{5}{x}^{4}-6300\,B{c}^{2}d{e}^{4}{x}^{4}-7280\,A{c}^{2}d{e}^{4}{x}^{3}+20020\,Bac{e}^{5}{x}^{3}+5600\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+25740\,Aac{e}^{5}{x}^{2}+6240\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-17160\,Bacd{e}^{4}{x}^{2}-4800\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-20592\,Aacd{e}^{4}x-4992\,A{c}^{2}{d}^{3}{e}^{2}x+18018\,B{a}^{2}{e}^{5}x+13728\,Bac{d}^{2}{e}^{3}x+3840\,B{c}^{2}{d}^{4}ex+30030\,A{a}^{2}{e}^{5}+13728\,A{d}^{2}ac{e}^{3}+3328\,A{d}^{4}{c}^{2}e-12012\,B{a}^{2}d{e}^{4}-9152\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2*(e*x+d)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3465*B*c^2*e^5*x^5+4095*A*c^2*e^5*x^4-3150*B*c^2*d*e^4*x^4-3640*A*c^2*d*e^4*x^3+10010*B
*a*c*e^5*x^3+2800*B*c^2*d^2*e^3*x^3+12870*A*a*c*e^5*x^2+3120*A*c^2*d^2*e^3*x^2-8580*B*a*c*d*e^4*x^2-2400*B*c^2
*d^3*e^2*x^2-10296*A*a*c*d*e^4*x-2496*A*c^2*d^3*e^2*x+9009*B*a^2*e^5*x+6864*B*a*c*d^2*e^3*x+1920*B*c^2*d^4*e*x
+15015*A*a^2*e^5+6864*A*a*c*d^2*e^3+1664*A*c^2*d^4*e-6006*B*a^2*d*e^4-4576*B*a*c*d^3*e^2-1280*B*c^2*d^5)/e^6

________________________________________________________________________________________

Maxima [A]  time = 1.02555, size = 335, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} B c^{2} - 4095 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*B*c^2 - 4095*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(11/2) + 10010*(5*B*c^2*d^2 - 2*A*
c^2*d*e + B*a*c*e^2)*(e*x + d)^(9/2) - 12870*(5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(e*x +
d)^(7/2) + 9009*(5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*(e*x + d)^(5/2) -
15015*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*(e*x + d)^(3/2))
/e^6

________________________________________________________________________________________

Fricas [A]  time = 1.48201, size = 774, normalized size = 3.55 \begin{align*} \frac{2 \,{\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 1664 \, A c^{2} d^{5} e - 4576 \, B a c d^{4} e^{2} + 6864 \, A a c d^{3} e^{3} - 6006 \, B a^{2} d^{2} e^{4} + 15015 \, A a^{2} d e^{5} + 315 \,{\left (B c^{2} d e^{5} + 13 \, A c^{2} e^{6}\right )} x^{5} - 35 \,{\left (10 \, B c^{2} d^{2} e^{4} - 13 \, A c^{2} d e^{5} - 286 \, B a c e^{6}\right )} x^{4} + 10 \,{\left (40 \, B c^{2} d^{3} e^{3} - 52 \, A c^{2} d^{2} e^{4} + 143 \, B a c d e^{5} + 1287 \, A a c e^{6}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{4} e^{2} - 208 \, A c^{2} d^{3} e^{3} + 572 \, B a c d^{2} e^{4} - 858 \, A a c d e^{5} - 3003 \, B a^{2} e^{6}\right )} x^{2} +{\left (640 \, B c^{2} d^{5} e - 832 \, A c^{2} d^{4} e^{2} + 2288 \, B a c d^{3} e^{3} - 3432 \, A a c d^{2} e^{4} + 3003 \, B a^{2} d e^{5} + 15015 \, A a^{2} e^{6}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/45045*(3465*B*c^2*e^6*x^6 - 1280*B*c^2*d^6 + 1664*A*c^2*d^5*e - 4576*B*a*c*d^4*e^2 + 6864*A*a*c*d^3*e^3 - 60
06*B*a^2*d^2*e^4 + 15015*A*a^2*d*e^5 + 315*(B*c^2*d*e^5 + 13*A*c^2*e^6)*x^5 - 35*(10*B*c^2*d^2*e^4 - 13*A*c^2*
d*e^5 - 286*B*a*c*e^6)*x^4 + 10*(40*B*c^2*d^3*e^3 - 52*A*c^2*d^2*e^4 + 143*B*a*c*d*e^5 + 1287*A*a*c*e^6)*x^3 -
3*(160*B*c^2*d^4*e^2 - 208*A*c^2*d^3*e^3 + 572*B*a*c*d^2*e^4 - 858*A*a*c*d*e^5 - 3003*B*a^2*e^6)*x^2 + (640*B
*c^2*d^5*e - 832*A*c^2*d^4*e^2 + 2288*B*a*c*d^3*e^3 - 3432*A*a*c*d^2*e^4 + 3003*B*a^2*d*e^5 + 15015*A*a^2*e^6)
*x)*sqrt(e*x + d)/e^6

________________________________________________________________________________________

Sympy [A]  time = 4.34836, size = 308, normalized size = 1.41 \begin{align*} \frac{2 \left (\frac{B c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 4 A a c d e^{3} - 4 A c^{2} d^{3} e + B a^{2} e^{4} + 6 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{2} e^{5} + 2 A a c d^{2} e^{3} + A c^{2} d^{4} e - B a^{2} d e^{4} - 2 B a c d^{3} e^{2} - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2*(e*x+d)**(1/2),x)

[Out]

2*(B*c**2*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*c**2*e - 5*B*c**2*d)/(11*e**5) + (d + e*x)**(9/2)
*(-4*A*c**2*d*e + 2*B*a*c*e**2 + 10*B*c**2*d**2)/(9*e**5) + (d + e*x)**(7/2)*(2*A*a*c*e**3 + 6*A*c**2*d**2*e -
6*B*a*c*d*e**2 - 10*B*c**2*d**3)/(7*e**5) + (d + e*x)**(5/2)*(-4*A*a*c*d*e**3 - 4*A*c**2*d**3*e + B*a**2*e**4
+ 6*B*a*c*d**2*e**2 + 5*B*c**2*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*a**2*e**5 + 2*A*a*c*d**2*e**3 + A*c**2*d*
*4*e - B*a**2*d*e**4 - 2*B*a*c*d**3*e**2 - B*c**2*d**5)/(3*e**5))/e

________________________________________________________________________________________

Giac [A]  time = 1.18048, size = 401, normalized size = 1.84 \begin{align*} \frac{2}{45045} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{2} e^{\left (-1\right )} + 858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} A a c e^{\left (-2\right )} + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} B a c e^{\left (-3\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} A c^{2} e^{\left (-4\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} B c^{2} e^{\left (-5\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^2*e^(-1) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^
(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*c*e^(-2) + 286*(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*a*c*e^(-3) + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 297
0*(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*e^(-4) + 5*(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*e^(-5) + 15015*(x*e + d)^(3/2)*A*a^2)*e^(-1)