3.588 \(\int (d+e x)^{3/2} \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=127 \[ \frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5} \]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)
^(7/2))/(7*e^5) + (4*c*(3*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d
+ e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.136245, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)*(a + c*x^2)^2,x]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)
^(7/2))/(7*e^5) + (4*c*(3*c*d^2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d
+ e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

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Rubi in Sympy [A]  time = 25.752, size = 122, normalized size = 0.96 \[ - \frac{8 c^{2} d \left (d + e x\right )^{\frac{11}{2}}}{11 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} - \frac{8 c d \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} + c d^{2}\right )}{7 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} + 3 c d^{2}\right )}{9 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(c*x**2+a)**2,x)

[Out]

-8*c**2*d*(d + e*x)**(11/2)/(11*e**5) + 2*c**2*(d + e*x)**(13/2)/(13*e**5) - 8*c
*d*(d + e*x)**(7/2)*(a*e**2 + c*d**2)/(7*e**5) + 4*c*(d + e*x)**(9/2)*(a*e**2 +
3*c*d**2)/(9*e**5) + 2*(d + e*x)**(5/2)*(a*e**2 + c*d**2)**2/(5*e**5)

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Mathematica [A]  time = 0.114722, size = 97, normalized size = 0.76 \[ \frac{2 (d+e x)^{5/2} \left (9009 a^2 e^4+286 a c e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 c^2 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(a + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(9009*a^2*e^4 + 286*a*c*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) +
3*c^2*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4))
)/(45045*e^5)

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Maple [A]  time = 0.01, size = 106, normalized size = 0.8 \[{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}-5040\,{c}^{2}d{x}^{3}{e}^{3}+20020\,ac{e}^{4}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-11440\,acd{e}^{3}x-1920\,{c}^{2}{d}^{3}ex+18018\,{a}^{2}{e}^{4}+4576\,ac{d}^{2}{e}^{2}+768\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(c*x^2+a)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4-2520*c^2*d*e^3*x^3+10010*a*c*e^4*x^2+168
0*c^2*d^2*e^2*x^2-5720*a*c*d*e^3*x-960*c^2*d^3*e*x+9009*a^2*e^4+2288*a*c*d^2*e^2
+384*c^2*d^4)/e^5

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Maxima [A]  time = 0.704235, size = 153, normalized size = 1.2 \[ \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 16380 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} d + 10010 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 25740 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 16380*(e*x + d)^(11/2)*c^2*d + 10010*(3*c^2
*d^2 + a*c*e^2)*(e*x + d)^(9/2) - 25740*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(7/2) +
9009*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(5/2))/e^5

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Fricas [A]  time = 0.232856, size = 244, normalized size = 1.92 \[ \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 4410 \, c^{2} d e^{5} x^{5} + 384 \, c^{2} d^{6} + 2288 \, a c d^{4} e^{2} + 9009 \, a^{2} d^{2} e^{4} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 286 \, a c e^{6}\right )} x^{4} - 20 \,{\left (6 \, c^{2} d^{3} e^{3} - 715 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} + 286 \, a c d^{2} e^{4} + 3003 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{5} e + 572 \, a c d^{3} e^{3} - 9009 \, a^{2} d e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

2/45045*(3465*c^2*e^6*x^6 + 4410*c^2*d*e^5*x^5 + 384*c^2*d^6 + 2288*a*c*d^4*e^2
+ 9009*a^2*d^2*e^4 + 35*(3*c^2*d^2*e^4 + 286*a*c*e^6)*x^4 - 20*(6*c^2*d^3*e^3 -
715*a*c*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2 + 286*a*c*d^2*e^4 + 3003*a^2*e^6)*x^2 - 2
*(96*c^2*d^5*e + 572*a*c*d^3*e^3 - 9009*a^2*d*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 6.28095, size = 328, normalized size = 2.58 \[ a^{2} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{4 a c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 c^{2} d \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{5}} + \frac{2 c^{2} \left (- \frac{d^{5} \left (d + e x\right )^{\frac{3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac{5}{2}} - \frac{10 d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{10 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9} - \frac{5 d \left (d + e x\right )^{\frac{11}{2}}}{11} + \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+a)**2,x)

[Out]

a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a*
*2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 4*a*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**3 + 4*a*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**3 + 2*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**5 + 2
*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**5

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GIAC/XCAS [A]  time = 0.21687, size = 443, normalized size = 3.49 \[ \frac{2}{45045} \,{\left (858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} a c d e^{\left (-14\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c^{2} d e^{\left (-44\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} a c e^{\left (-26\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{60} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{60} + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{60} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{60} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{60} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{60}\right )} c^{2} e^{\left (-64\right )} + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(e*x + d)^(3/2),x, algorithm="giac")

[Out]

2/45045*(858*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)
^(3/2)*d^2*e^12)*a*c*d*e^(-14) + 13*(315*(x*e + d)^(11/2)*e^40 - 1540*(x*e + d)^
(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3*e^40 + 1
155*(x*e + d)^(3/2)*d^4*e^40)*c^2*d*e^(-44) + 15015*(x*e + d)^(3/2)*a^2*d + 286*
(35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*
e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*a*c*e^(-26) + 5*(693*(x*e + d)^(13/2)*e^60
- 4095*(x*e + d)^(11/2)*d*e^60 + 10010*(x*e + d)^(9/2)*d^2*e^60 - 12870*(x*e + d
)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(5/2)*d^4*e^60 - 3003*(x*e + d)^(3/2)*d^5*e^60
)*c^2*e^(-64) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2)*e^(-1)