3.587 \(\int (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=127 \[ \frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5} \]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)
^(9/2))/(9*e^5) + (4*c*(3*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(
d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5)

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Rubi [A]  time = 0.150057, 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 \[ \frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)
^(9/2))/(9*e^5) + (4*c*(3*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(
d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5)

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Rubi in Sympy [A]  time = 26.1401, size = 122, normalized size = 0.96 \[ - \frac{8 c^{2} d \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{5}} - \frac{8 c d \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} + c d^{2}\right )}{9 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{11}{2}} \left (a e^{2} + 3 c d^{2}\right )}{11 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} + c d^{2}\right )^{2}}{7 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*c**2*d*(d + e*x)**(13/2)/(13*e**5) + 2*c**2*(d + e*x)**(15/2)/(15*e**5) - 8*c
*d*(d + e*x)**(9/2)*(a*e**2 + c*d**2)/(9*e**5) + 4*c*(d + e*x)**(11/2)*(a*e**2 +
3*c*d**2)/(11*e**5) + 2*(d + e*x)**(7/2)*(a*e**2 + c*d**2)**2/(7*e**5)

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Mathematica [A]  time = 0.127338, size = 96, normalized size = 0.76 \[ \frac{2 (d+e x)^{7/2} \left (6435 a^2 e^4+130 a c e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d + e*x)^(7/2)*(6435*a^2*e^4 + 130*a*c*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) +
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))
)/(45045*e^5)

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Maple [A]  time = 0.01, size = 106, normalized size = 0.8 \[{\frac{6006\,{c}^{2}{x}^{4}{e}^{4}-3696\,{c}^{2}d{x}^{3}{e}^{3}+16380\,ac{e}^{4}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-7280\,acd{e}^{3}x-896\,{c}^{2}{d}^{3}ex+12870\,{a}^{2}{e}^{4}+2080\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4-1848*c^2*d*e^3*x^3+8190*a*c*e^4*x^2+1008
*c^2*d^2*e^2*x^2-3640*a*c*d*e^3*x-448*c^2*d^3*e*x+6435*a^2*e^4+1040*a*c*d^2*e^2+
128*c^2*d^4)/e^5

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Maxima [A]  time = 0.701343, size = 153, normalized size = 1.2 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 13860 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} d + 8190 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 13860*(e*x + d)^(13/2)*c^2*d + 8190*(3*c^2*
d^2 + a*c*e^2)*(e*x + d)^(11/2) - 20020*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(9/2) +
6435*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(7/2))/e^5

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Fricas [A]  time = 0.236571, size = 294, normalized size = 2.31 \[ \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 7161 \, c^{2} d e^{6} x^{6} + 128 \, c^{2} d^{7} + 1040 \, a c d^{5} e^{2} + 6435 \, a^{2} d^{3} e^{4} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 130 \, a c e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 598 \, a c d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 2938 \, a c d^{2} e^{5} - 1287 \, a^{2} e^{7}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} + 130 \, a c d^{3} e^{4} + 6435 \, a^{2} d e^{6}\right )} x^{2} -{\left (64 \, c^{2} d^{6} e + 520 \, a c d^{4} e^{3} - 19305 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*c^2*e^7*x^7 + 7161*c^2*d*e^6*x^6 + 128*c^2*d^7 + 1040*a*c*d^5*e^2
+ 6435*a^2*d^3*e^4 + 63*(71*c^2*d^2*e^5 + 130*a*c*e^7)*x^5 + 35*(c^2*d^3*e^4 + 5
98*a*c*d*e^6)*x^4 - 5*(8*c^2*d^4*e^3 - 2938*a*c*d^2*e^5 - 1287*a^2*e^7)*x^3 + 3*
(16*c^2*d^5*e^2 + 130*a*c*d^3*e^4 + 6435*a^2*d*e^6)*x^2 - (64*c^2*d^6*e + 520*a*
c*d^4*e^3 - 19305*a^2*d^2*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 9.54119, size = 566, normalized size = 4.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4
*a**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**2*(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*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 + 8*a*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 + 4*a*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 + 2*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*
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*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)/1
1 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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GIAC/XCAS [A]  time = 0.220534, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done