3.594 \(\int (d+e x)^{5/2} \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=204 \[ \frac{2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac{6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7} \]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*
x)^(9/2))/(3*e^7) + (6*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11
*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c^2*(5*c*d^
2 + a*e^2)*(d + e*x)^(15/2))/(5*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) + (2
*c^3*(d + e*x)^(19/2))/(19*e^7)

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Rubi [A]  time = 0.25859, antiderivative size = 204, 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 \[ \frac{2 c^2 (d+e x)^{15/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac{8 c^2 d (d+e x)^{13/2} \left (3 a e^2+5 c d^2\right )}{13 e^7}+\frac{6 c (d+e x)^{11/2} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{11 e^7}-\frac{4 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )^2}{3 e^7}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}-\frac{12 c^3 d (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)*(a + c*x^2)^3,x]

[Out]

(2*(c*d^2 + a*e^2)^3*(d + e*x)^(7/2))/(7*e^7) - (4*c*d*(c*d^2 + a*e^2)^2*(d + e*
x)^(9/2))/(3*e^7) + (6*c*(c*d^2 + a*e^2)*(5*c*d^2 + a*e^2)*(d + e*x)^(11/2))/(11
*e^7) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c^2*(5*c*d^
2 + a*e^2)*(d + e*x)^(15/2))/(5*e^7) - (12*c^3*d*(d + e*x)^(17/2))/(17*e^7) + (2
*c^3*(d + e*x)^(19/2))/(19*e^7)

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Rubi in Sympy [A]  time = 44.5764, size = 199, normalized size = 0.98 \[ - \frac{12 c^{3} d \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{2 c^{3} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{7}} - \frac{8 c^{2} d \left (d + e x\right )^{\frac{13}{2}} \left (3 a e^{2} + 5 c d^{2}\right )}{13 e^{7}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}} \left (a e^{2} + 5 c d^{2}\right )}{5 e^{7}} - \frac{4 c d \left (d + e x\right )^{\frac{9}{2}} \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{7}} + \frac{6 c \left (d + e x\right )^{\frac{11}{2}} \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{11 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} + c d^{2}\right )^{3}}{7 e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)*(c*x**2+a)**3,x)

[Out]

-12*c**3*d*(d + e*x)**(17/2)/(17*e**7) + 2*c**3*(d + e*x)**(19/2)/(19*e**7) - 8*
c**2*d*(d + e*x)**(13/2)*(3*a*e**2 + 5*c*d**2)/(13*e**7) + 2*c**2*(d + e*x)**(15
/2)*(a*e**2 + 5*c*d**2)/(5*e**7) - 4*c*d*(d + e*x)**(9/2)*(a*e**2 + c*d**2)**2/(
3*e**7) + 6*c*(d + e*x)**(11/2)*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)/(11*e**7)
+ 2*(d + e*x)**(7/2)*(a*e**2 + c*d**2)**3/(7*e**7)

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Mathematica [A]  time = 0.185675, size = 171, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (692835 a^3 e^6+20995 a^2 c e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+323 a c^2 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+5 c^3 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{4849845 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)*(a + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(7/2)*(692835*a^3*e^6 + 20995*a^2*c*e^4*(8*d^2 - 28*d*e*x + 63*e^2*
x^2) + 323*a*c^2*e^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) + 5*c^3*(1024*d^6 - 3584*d^5*e*x + 8064*d^4*e^2*x^2 - 14784*d^3*
e^3*x^3 + 24024*d^2*e^4*x^4 - 36036*d*e^5*x^5 + 51051*e^6*x^6)))/(4849845*e^7)

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Maple [A]  time = 0.01, size = 205, normalized size = 1. \[{\frac{510510\,{c}^{3}{x}^{6}{e}^{6}-360360\,{c}^{3}d{x}^{5}{e}^{5}+1939938\,a{c}^{2}{e}^{6}{x}^{4}+240240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-1193808\,a{c}^{2}d{e}^{5}{x}^{3}-147840\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+2645370\,{a}^{2}c{e}^{6}{x}^{2}+651168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+80640\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-1175720\,{a}^{2}cd{e}^{5}x-289408\,a{c}^{2}{d}^{3}{e}^{3}x-35840\,{c}^{3}{d}^{5}ex+1385670\,{a}^{3}{e}^{6}+335920\,{a}^{2}c{d}^{2}{e}^{4}+82688\,a{c}^{2}{d}^{4}{e}^{2}+10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)*(c*x^2+a)^3,x)

[Out]

2/4849845*(e*x+d)^(7/2)*(255255*c^3*e^6*x^6-180180*c^3*d*e^5*x^5+969969*a*c^2*e^
6*x^4+120120*c^3*d^2*e^4*x^4-596904*a*c^2*d*e^5*x^3-73920*c^3*d^3*e^3*x^3+132268
5*a^2*c*e^6*x^2+325584*a*c^2*d^2*e^4*x^2+40320*c^3*d^4*e^2*x^2-587860*a^2*c*d*e^
5*x-144704*a*c^2*d^3*e^3*x-17920*c^3*d^5*e*x+692835*a^3*e^6+167960*a^2*c*d^2*e^4
+41344*a*c^2*d^4*e^2+5120*c^3*d^6)/e^7

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Maxima [A]  time = 0.706041, size = 282, normalized size = 1.38 \[ \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 1711710 \,{\left (e x + d\right )}^{\frac{17}{2}} c^{3} d + 969969 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1492260 \,{\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 3233230 \,{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 692835 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{4849845 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 1711710*(e*x + d)^(17/2)*c^3*d + 969969
*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(15/2) - 1492260*(5*c^3*d^3 + 3*a*c^2*d*e^2)*
(e*x + d)^(13/2) + 1322685*(5*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^(
11/2) - 3233230*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d)^(9/2) + 6928
35*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(e*x + d)^(7/2))/e^7

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Fricas [A]  time = 0.243502, size = 481, normalized size = 2.36 \[ \frac{2 \,{\left (255255 \, c^{3} e^{9} x^{9} + 585585 \, c^{3} d e^{8} x^{8} + 5120 \, c^{3} d^{9} + 41344 \, a c^{2} d^{7} e^{2} + 167960 \, a^{2} c d^{5} e^{4} + 692835 \, a^{3} d^{3} e^{6} + 3003 \,{\left (115 \, c^{3} d^{2} e^{7} + 323 \, a c^{2} e^{9}\right )} x^{7} + 231 \,{\left (5 \, c^{3} d^{3} e^{6} + 10013 \, a c^{2} d e^{8}\right )} x^{6} - 63 \,{\left (20 \, c^{3} d^{4} e^{5} - 22933 \, a c^{2} d^{2} e^{7} - 20995 \, a^{2} c e^{9}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{5} e^{4} + 323 \, a c^{2} d^{3} e^{6} + 96577 \, a^{2} c d e^{8}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{6} e^{3} + 2584 \, a c^{2} d^{4} e^{5} - 474487 \, a^{2} c d^{2} e^{7} - 138567 \, a^{3} e^{9}\right )} x^{3} + 3 \,{\left (640 \, c^{3} d^{7} e^{2} + 5168 \, a c^{2} d^{5} e^{4} + 20995 \, a^{2} c d^{3} e^{6} + 692835 \, a^{3} d e^{8}\right )} x^{2} -{\left (2560 \, c^{3} d^{8} e + 20672 \, a c^{2} d^{6} e^{3} + 83980 \, a^{2} c d^{4} e^{5} - 2078505 \, a^{3} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{4849845 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 585585*c^3*d*e^8*x^8 + 5120*c^3*d^9 + 41344*a*c^
2*d^7*e^2 + 167960*a^2*c*d^5*e^4 + 692835*a^3*d^3*e^6 + 3003*(115*c^3*d^2*e^7 +
323*a*c^2*e^9)*x^7 + 231*(5*c^3*d^3*e^6 + 10013*a*c^2*d*e^8)*x^6 - 63*(20*c^3*d^
4*e^5 - 22933*a*c^2*d^2*e^7 - 20995*a^2*c*e^9)*x^5 + 35*(40*c^3*d^5*e^4 + 323*a*
c^2*d^3*e^6 + 96577*a^2*c*d*e^8)*x^4 - 5*(320*c^3*d^6*e^3 + 2584*a*c^2*d^4*e^5 -
474487*a^2*c*d^2*e^7 - 138567*a^3*e^9)*x^3 + 3*(640*c^3*d^7*e^2 + 5168*a*c^2*d^
5*e^4 + 20995*a^2*c*d^3*e^6 + 692835*a^3*d*e^8)*x^2 - (2560*c^3*d^8*e + 20672*a*
c^2*d^6*e^3 + 83980*a^2*c*d^4*e^5 - 2078505*a^3*d^2*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [A]  time = 12.4227, size = 945, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)*(c*x**2+a)**3,x)

[Out]

a**3*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4
*a**3*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e + 2*a**3*(d**2*(d + e*x)*
*(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e + 6*a**2*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 + 12*a**2
*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 + 6*a**2*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 + 6*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)/1
1)/e**5 + 12*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 + 6*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*c
**3*d**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**7 + 4*c**3*d*(-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**7 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 -
8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/
9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e
*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7

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GIAC/XCAS [A]  time = 0.228381, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Done