3.238 \(\int x^5 \sqrt{-c+d x} \sqrt{c+d x} \left (a+b x^2\right ) \, dx\)

Optimal. Leaf size=152 \[ \frac{(d x-c)^{7/2} (c+d x)^{7/2} \left (a d^2+3 b c^2\right )}{7 d^8}+\frac{c^2 (d x-c)^{5/2} (c+d x)^{5/2} \left (2 a d^2+3 b c^2\right )}{5 d^8}+\frac{c^4 (d x-c)^{3/2} (c+d x)^{3/2} \left (a d^2+b c^2\right )}{3 d^8}+\frac{b (d x-c)^{9/2} (c+d x)^{9/2}}{9 d^8} \]

[Out]

(c^4*(b*c^2 + a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^8) + (c^2*(3*b*c^2 +
2*a*d^2)*(-c + d*x)^(5/2)*(c + d*x)^(5/2))/(5*d^8) + ((3*b*c^2 + a*d^2)*(-c + d
*x)^(7/2)*(c + d*x)^(7/2))/(7*d^8) + (b*(-c + d*x)^(9/2)*(c + d*x)^(9/2))/(9*d^8
)

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Rubi [A]  time = 0.406873, antiderivative size = 164, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{4 c^2 x^2 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{105 d^6}+\frac{x^4 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{21 d^4}+\frac{8 c^4 (d x-c)^{3/2} (c+d x)^{3/2} \left (3 a d^2+2 b c^2\right )}{315 d^8}+\frac{b x^6 (d x-c)^{3/2} (c+d x)^{3/2}}{9 d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

(8*c^4*(2*b*c^2 + 3*a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(315*d^8) + (4*c^2*
(2*b*c^2 + 3*a*d^2)*x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(105*d^6) + ((2*b*c^2
+ 3*a*d^2)*x^4*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(21*d^4) + (b*x^6*(-c + d*x)^(3
/2)*(c + d*x)^(3/2))/(9*d^2)

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Rubi in Sympy [A]  time = 25.8529, size = 150, normalized size = 0.99 \[ \frac{b x^{6} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}{9 d^{2}} + \frac{8 c^{4} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d^{2} + 2 b c^{2}\right )}{315 d^{8}} + \frac{4 c^{2} x^{2} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d^{2} + 2 b c^{2}\right )}{105 d^{6}} + \frac{x^{4} \left (- c + d x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}} \left (3 a d^{2} + 2 b c^{2}\right )}{21 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2),x)

[Out]

b*x**6*(-c + d*x)**(3/2)*(c + d*x)**(3/2)/(9*d**2) + 8*c**4*(-c + d*x)**(3/2)*(c
+ d*x)**(3/2)*(3*a*d**2 + 2*b*c**2)/(315*d**8) + 4*c**2*x**2*(-c + d*x)**(3/2)*
(c + d*x)**(3/2)*(3*a*d**2 + 2*b*c**2)/(105*d**6) + x**4*(-c + d*x)**(3/2)*(c +
d*x)**(3/2)*(3*a*d**2 + 2*b*c**2)/(21*d**4)

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Mathematica [A]  time = 0.094186, size = 109, normalized size = 0.72 \[ -\frac{\sqrt{d x-c} \sqrt{c+d x} \left (c^2-d^2 x^2\right ) \left (3 a d^2 \left (8 c^4+12 c^2 d^2 x^2+15 d^4 x^4\right )+b \left (16 c^6+24 c^4 d^2 x^2+30 c^2 d^4 x^4+35 d^6 x^6\right )\right )}{315 d^8} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2),x]

[Out]

-(Sqrt[-c + d*x]*Sqrt[c + d*x]*(c^2 - d^2*x^2)*(3*a*d^2*(8*c^4 + 12*c^2*d^2*x^2
+ 15*d^4*x^4) + b*(16*c^6 + 24*c^4*d^2*x^2 + 30*c^2*d^4*x^4 + 35*d^6*x^6)))/(315
*d^8)

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Maple [A]  time = 0.01, size = 92, normalized size = 0.6 \[{\frac{35\,b{x}^{6}{d}^{6}+45\,a{d}^{6}{x}^{4}+30\,b{c}^{2}{d}^{4}{x}^{4}+36\,a{c}^{2}{d}^{4}{x}^{2}+24\,b{c}^{4}{d}^{2}{x}^{2}+24\,a{c}^{4}{d}^{2}+16\,b{c}^{6}}{315\,{d}^{8}} \left ( dx+c \right ) ^{{\frac{3}{2}}} \left ( dx-c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)*(d*x-c)^(1/2)*(d*x+c)^(1/2),x)

[Out]

1/315*(d*x+c)^(3/2)*(35*b*d^6*x^6+45*a*d^6*x^4+30*b*c^2*d^4*x^4+36*a*c^2*d^4*x^2
+24*b*c^4*d^2*x^2+24*a*c^4*d^2+16*b*c^6)*(d*x-c)^(3/2)/d^8

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Maxima [A]  time = 1.39231, size = 240, normalized size = 1.58 \[ \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b x^{6}}{9 \, d^{2}} + \frac{2 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{2} x^{4}}{21 \, d^{4}} + \frac{{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a x^{4}}{7 \, d^{2}} + \frac{8 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{4} x^{2}}{105 \, d^{6}} + \frac{4 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{2} x^{2}}{35 \, d^{4}} + \frac{16 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} b c^{6}}{315 \, d^{8}} + \frac{8 \,{\left (d^{2} x^{2} - c^{2}\right )}^{\frac{3}{2}} a c^{4}}{105 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^5,x, algorithm="maxima")

[Out]

1/9*(d^2*x^2 - c^2)^(3/2)*b*x^6/d^2 + 2/21*(d^2*x^2 - c^2)^(3/2)*b*c^2*x^4/d^4 +
1/7*(d^2*x^2 - c^2)^(3/2)*a*x^4/d^2 + 8/105*(d^2*x^2 - c^2)^(3/2)*b*c^4*x^2/d^6
+ 4/35*(d^2*x^2 - c^2)^(3/2)*a*c^2*x^2/d^4 + 16/315*(d^2*x^2 - c^2)^(3/2)*b*c^6
/d^8 + 8/105*(d^2*x^2 - c^2)^(3/2)*a*c^4/d^6

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Fricas [A]  time = 0.743305, size = 741, normalized size = 4.88 \[ -\frac{8960 \, b d^{18} x^{18} + 16 \, b c^{18} + 24 \, a c^{16} d^{2} - 2880 \,{\left (9 \, b c^{2} d^{16} - 4 \, a d^{18}\right )} x^{16} + 144 \,{\left (181 \, b c^{4} d^{14} - 236 \, a c^{2} d^{16}\right )} x^{14} - 24 \,{\left (461 \, b c^{6} d^{12} - 1426 \, a c^{4} d^{14}\right )} x^{12} + 9 \,{\left (27 \, b c^{8} d^{10} - 1832 \, a c^{6} d^{12}\right )} x^{10} + 45 \,{\left (160 \, b c^{10} d^{8} + 289 \, a c^{8} d^{10}\right )} x^{8} - 21 \,{\left (429 \, b c^{12} d^{6} + 646 \, a c^{10} d^{8}\right )} x^{6} + 2079 \,{\left (2 \, b c^{14} d^{4} + 3 \, a c^{12} d^{6}\right )} x^{4} - 324 \,{\left (2 \, b c^{16} d^{2} + 3 \, a c^{14} d^{4}\right )} x^{2} -{\left (8960 \, b d^{17} x^{17} - 320 \,{\left (67 \, b c^{2} d^{15} - 36 \, a d^{17}\right )} x^{15} + 2352 \,{\left (7 \, b c^{4} d^{13} - 12 \, a c^{2} d^{15}\right )} x^{13} - 8 \,{\left (619 \, b c^{6} d^{11} - 2694 \, a c^{4} d^{13}\right )} x^{11} - 5 \,{\left (233 \, b c^{8} d^{9} + 1704 \, a c^{6} d^{11}\right )} x^{9} + 45 \,{\left (143 \, b c^{10} d^{7} + 225 \, a c^{8} d^{9}\right )} x^{7} - 3003 \,{\left (2 \, b c^{12} d^{5} + 3 \, a c^{10} d^{7}\right )} x^{5} + 924 \,{\left (2 \, b c^{14} d^{3} + 3 \, a c^{12} d^{5}\right )} x^{3} - 72 \,{\left (2 \, b c^{16} d + 3 \, a c^{14} d^{3}\right )} x\right )} \sqrt{d x + c} \sqrt{d x - c}}{315 \,{\left (256 \, d^{17} x^{9} - 576 \, c^{2} d^{15} x^{7} + 432 \, c^{4} d^{13} x^{5} - 120 \, c^{6} d^{11} x^{3} + 9 \, c^{8} d^{9} x -{\left (256 \, d^{16} x^{8} - 448 \, c^{2} d^{14} x^{6} + 240 \, c^{4} d^{12} x^{4} - 40 \, c^{6} d^{10} x^{2} + c^{8} d^{8}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^5,x, algorithm="fricas")

[Out]

-1/315*(8960*b*d^18*x^18 + 16*b*c^18 + 24*a*c^16*d^2 - 2880*(9*b*c^2*d^16 - 4*a*
d^18)*x^16 + 144*(181*b*c^4*d^14 - 236*a*c^2*d^16)*x^14 - 24*(461*b*c^6*d^12 - 1
426*a*c^4*d^14)*x^12 + 9*(27*b*c^8*d^10 - 1832*a*c^6*d^12)*x^10 + 45*(160*b*c^10
*d^8 + 289*a*c^8*d^10)*x^8 - 21*(429*b*c^12*d^6 + 646*a*c^10*d^8)*x^6 + 2079*(2*
b*c^14*d^4 + 3*a*c^12*d^6)*x^4 - 324*(2*b*c^16*d^2 + 3*a*c^14*d^4)*x^2 - (8960*b
*d^17*x^17 - 320*(67*b*c^2*d^15 - 36*a*d^17)*x^15 + 2352*(7*b*c^4*d^13 - 12*a*c^
2*d^15)*x^13 - 8*(619*b*c^6*d^11 - 2694*a*c^4*d^13)*x^11 - 5*(233*b*c^8*d^9 + 17
04*a*c^6*d^11)*x^9 + 45*(143*b*c^10*d^7 + 225*a*c^8*d^9)*x^7 - 3003*(2*b*c^12*d^
5 + 3*a*c^10*d^7)*x^5 + 924*(2*b*c^14*d^3 + 3*a*c^12*d^5)*x^3 - 72*(2*b*c^16*d +
3*a*c^14*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c))/(256*d^17*x^9 - 576*c^2*d^15*x^7
+ 432*c^4*d^13*x^5 - 120*c^6*d^11*x^3 + 9*c^8*d^9*x - (256*d^16*x^8 - 448*c^2*d^
14*x^6 + 240*c^4*d^12*x^4 - 40*c^6*d^10*x^2 + c^8*d^8)*sqrt(d*x + c)*sqrt(d*x -
c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \left (a + b x^{2}\right ) \sqrt{- c + d x} \sqrt{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)*(d*x-c)**(1/2)*(d*x+c)**(1/2),x)

[Out]

Integral(x**5*(a + b*x**2)*sqrt(-c + d*x)*sqrt(c + d*x), x)

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GIAC/XCAS [A]  time = 0.257139, size = 309, normalized size = 2.03 \[ \frac{3 \,{\left ({\left (3 \,{\left ({\left (d x + c\right )}{\left (5 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{5}} - \frac{6 \, c}{d^{5}}\right )} + \frac{74 \, c^{2}}{d^{5}}\right )} - \frac{96 \, c^{3}}{d^{5}}\right )}{\left (d x + c\right )} + \frac{203 \, c^{4}}{d^{5}}\right )}{\left (d x + c\right )} - \frac{70 \, c^{5}}{d^{5}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} a +{\left ({\left ({\left ({\left (5 \,{\left ({\left (d x + c\right )}{\left (7 \,{\left (d x + c\right )}{\left (\frac{d x + c}{d^{7}} - \frac{8 \, c}{d^{7}}\right )} + \frac{195 \, c^{2}}{d^{7}}\right )} - \frac{386 \, c^{3}}{d^{7}}\right )}{\left (d x + c\right )} + \frac{2369 \, c^{4}}{d^{7}}\right )}{\left (d x + c\right )} - \frac{1836 \, c^{5}}{d^{7}}\right )}{\left (d x + c\right )} + \frac{861 \, c^{6}}{d^{7}}\right )}{\left (d x + c\right )} - \frac{210 \, c^{7}}{d^{7}}\right )}{\left (d x + c\right )}^{\frac{3}{2}} \sqrt{d x - c} b}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*sqrt(d*x + c)*sqrt(d*x - c)*x^5,x, algorithm="giac")

[Out]

1/315*(3*((3*((d*x + c)*(5*(d*x + c)*((d*x + c)/d^5 - 6*c/d^5) + 74*c^2/d^5) - 9
6*c^3/d^5)*(d*x + c) + 203*c^4/d^5)*(d*x + c) - 70*c^5/d^5)*(d*x + c)^(3/2)*sqrt
(d*x - c)*a + ((((5*((d*x + c)*(7*(d*x + c)*((d*x + c)/d^7 - 8*c/d^7) + 195*c^2/
d^7) - 386*c^3/d^7)*(d*x + c) + 2369*c^4/d^7)*(d*x + c) - 1836*c^5/d^7)*(d*x + c
) + 861*c^6/d^7)*(d*x + c) - 210*c^7/d^7)*(d*x + c)^(3/2)*sqrt(d*x - c)*b)/d