3.1439 \(\int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=214 \[ \frac{4 c \sqrt{d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(5*e^6*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)*(
5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^(3/2)) + (4*c*(5*B*c*d^3 - 3*
A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*Sqrt[d + e*x]) + (4*c*(5*B*c*d^2 - 2*A*
c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^6 - (2*c^2*(5*B*d - A*e)*(d + e*x)^(3/2))/(3*e
^6) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6)

_______________________________________________________________________________________

Rubi [A]  time = 0.252536, antiderivative size = 214, 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 \[ \frac{4 c \sqrt{d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

(2*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(5*e^6*(d + e*x)^(5/2)) - (2*(c*d^2 + a*e^2)*(
5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^(3/2)) + (4*c*(5*B*c*d^3 - 3*
A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^6*Sqrt[d + e*x]) + (4*c*(5*B*c*d^2 - 2*A*
c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^6 - (2*c^2*(5*B*d - A*e)*(d + e*x)^(3/2))/(3*e
^6) + (2*B*c^2*(d + e*x)^(5/2))/(5*e^6)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.5958, size = 219, normalized size = 1.02 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{3}{2}} \left (A e - 5 B d\right )}{3 e^{6}} + \frac{4 c \sqrt{d + e x} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{e^{6}} - \frac{4 c \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{e^{6} \sqrt{d + e x}} - \frac{2 \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{3 e^{6} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{5 e^{6} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(7/2),x)

[Out]

2*B*c**2*(d + e*x)**(5/2)/(5*e**6) + 2*c**2*(d + e*x)**(3/2)*(A*e - 5*B*d)/(3*e*
*6) + 4*c*sqrt(d + e*x)*(-2*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/e**6 - 4*c*(A*a*e**
3 + 3*A*c*d**2*e - 3*B*a*d*e**2 - 5*B*c*d**3)/(e**6*sqrt(d + e*x)) - 2*(a*e**2 +
c*d**2)*(-4*A*c*d*e + B*a*e**2 + 5*B*c*d**2)/(3*e**6*(d + e*x)**(3/2)) - 2*(A*e
- B*d)*(a*e**2 + c*d**2)**2/(5*e**6*(d + e*x)**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.734104, size = 178, normalized size = 0.83 \[ \frac{2 \sqrt{d+e x} \left (-\frac{5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{(d+e x)^2}+\frac{3 \left (a e^2+c d^2\right )^2 (B d-A e)}{(d+e x)^3}+c \left (30 a B e^2-55 A c d e+128 B c d^2\right )+\frac{30 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{d+e x}+c^2 e x (5 A e-19 B d)+3 B c^2 e^2 x^2\right )}{15 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^(7/2),x]

[Out]

(2*Sqrt[d + e*x]*(c*(128*B*c*d^2 - 55*A*c*d*e + 30*a*B*e^2) + c^2*e*(-19*B*d + 5
*A*e)*x + 3*B*c^2*e^2*x^2 + (3*(B*d - A*e)*(c*d^2 + a*e^2)^2)/(d + e*x)^3 - (5*(
c*d^2 + a*e^2)*(5*B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(d + e*x)^2 + (30*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)))/(15*e^6)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 259, normalized size = 1.2 \[ -{\frac{-6\,B{c}^{2}{x}^{5}{e}^{5}-10\,A{c}^{2}{e}^{5}{x}^{4}+20\,B{c}^{2}d{e}^{4}{x}^{4}+80\,A{c}^{2}d{e}^{4}{x}^{3}-60\,Bac{e}^{5}{x}^{3}-160\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+60\,Aac{e}^{5}{x}^{2}+480\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-360\,Bacd{e}^{4}{x}^{2}-960\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}+80\,Aacd{e}^{4}x+640\,A{c}^{2}{d}^{3}{e}^{2}x+10\,B{a}^{2}{e}^{5}x-480\,Bac{d}^{2}{e}^{3}x-1280\,B{c}^{2}{d}^{4}ex+6\,A{a}^{2}{e}^{5}+32\,A{d}^{2}ac{e}^{3}+256\,A{d}^{4}{c}^{2}e+4\,Bd{a}^{2}{e}^{4}-192\,aBc{d}^{3}{e}^{2}-512\,B{c}^{2}{d}^{5}}{15\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^2/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-3*B*c^2*e^5*x^5-5*A*c^2*e^5*x^4+10*B*c^2*d*e^4*x^4+40*A*c^
2*d*e^4*x^3-30*B*a*c*e^5*x^3-80*B*c^2*d^2*e^3*x^3+30*A*a*c*e^5*x^2+240*A*c^2*d^2
*e^3*x^2-180*B*a*c*d*e^4*x^2-480*B*c^2*d^3*e^2*x^2+40*A*a*c*d*e^4*x+320*A*c^2*d^
3*e^2*x+5*B*a^2*e^5*x-240*B*a*c*d^2*e^3*x-640*B*c^2*d^4*e*x+3*A*a^2*e^5+16*A*a*c
*d^2*e^3+128*A*c^2*d^4*e+2*B*a^2*d*e^4-96*B*a*c*d^3*e^2-256*B*c^2*d^5)/e^6

_______________________________________________________________________________________

Maxima [A]  time = 0.687493, size = 344, normalized size = 1.61 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c^{2} - 5 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 30 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} \sqrt{e x + d}}{e^{5}} + \frac{3 \, B c^{2} d^{5} - 3 \, A c^{2} d^{4} e + 6 \, B a c d^{3} e^{2} - 6 \, A a c d^{2} e^{3} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} + 30 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{5}}\right )}}{15 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*B*c^2 - 5*(5*B*c^2*d - A*c^2*e)*(e*x + d)^(3/2) + 30*(5
*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*sqrt(e*x + d))/e^5 + (3*B*c^2*d^5 - 3*A*c^
2*d^4*e + 6*B*a*c*d^3*e^2 - 6*A*a*c*d^2*e^3 + 3*B*a^2*d*e^4 - 3*A*a^2*e^5 + 30*(
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)^2 - 5*(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))/
((e*x + d)^(5/2)*e^5))/e

_______________________________________________________________________________________

Fricas [A]  time = 0.268702, size = 363, normalized size = 1.7 \[ \frac{2 \,{\left (3 \, B c^{2} e^{5} x^{5} + 256 \, B c^{2} d^{5} - 128 \, A c^{2} d^{4} e + 96 \, B a c d^{3} e^{2} - 16 \, A a c d^{2} e^{3} - 2 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5} - 5 \,{\left (2 \, B c^{2} d e^{4} - A c^{2} e^{5}\right )} x^{4} + 10 \,{\left (8 \, B c^{2} d^{2} e^{3} - 4 \, A c^{2} d e^{4} + 3 \, B a c e^{5}\right )} x^{3} + 30 \,{\left (16 \, B c^{2} d^{3} e^{2} - 8 \, A c^{2} d^{2} e^{3} + 6 \, B a c d e^{4} - A a c e^{5}\right )} x^{2} + 5 \,{\left (128 \, B c^{2} d^{4} e - 64 \, A c^{2} d^{3} e^{2} + 48 \, B a c d^{2} e^{3} - 8 \, A a c d e^{4} - B a^{2} e^{5}\right )} x\right )}}{15 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )} \sqrt{e x + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(3*B*c^2*e^5*x^5 + 256*B*c^2*d^5 - 128*A*c^2*d^4*e + 96*B*a*c*d^3*e^2 - 16*
A*a*c*d^2*e^3 - 2*B*a^2*d*e^4 - 3*A*a^2*e^5 - 5*(2*B*c^2*d*e^4 - A*c^2*e^5)*x^4
+ 10*(8*B*c^2*d^2*e^3 - 4*A*c^2*d*e^4 + 3*B*a*c*e^5)*x^3 + 30*(16*B*c^2*d^3*e^2
- 8*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 - A*a*c*e^5)*x^2 + 5*(128*B*c^2*d^4*e - 64*A*c
^2*d^3*e^2 + 48*B*a*c*d^2*e^3 - 8*A*a*c*d*e^4 - B*a^2*e^5)*x)/((e^8*x^2 + 2*d*e^
7*x + d^2*e^6)*sqrt(e*x + d))

_______________________________________________________________________________________

Sympy [A]  time = 11.9933, size = 1426, normalized size = 6.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**2*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) - 32*A*a*c*d**2*e**3/(15*d**2*e**6*sqrt(d + e*x)
+ 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 80*A*a*c*d*e**4*x/(
15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e
*x)) - 60*A*a*c*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) - 256*A*c**2*d**4*e/(15*d**2*e**6*sqrt(d + e*x)
+ 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 640*A*c**2*d**3*e**2
*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d
+ e*x)) - 480*A*c**2*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*s
qrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 80*A*c**2*d*e**4*x**3/(15*d**2*e**6
*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 10*A*
c**2*e**5*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8
*x**2*sqrt(d + e*x)) - 4*B*a**2*d*e**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x
*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 10*B*a**2*e**5*x/(15*d**2*e**6*sq
rt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 192*B*a*
c*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x*
*2*sqrt(d + e*x)) + 480*B*a*c*d**2*e**3*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**
7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 360*B*a*c*d*e**4*x**2/(15*d**2
*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) +
60*B*a*c*e**5*x**3/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*
e**8*x**2*sqrt(d + e*x)) + 512*B*c**2*d**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e*
*7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1280*B*c**2*d**4*e*x/(15*d**2
*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) +
960*B*c**2*d**3*e**2*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x
) + 15*e**8*x**2*sqrt(d + e*x)) + 160*B*c**2*d**2*e**3*x**3/(15*d**2*e**6*sqrt(d
+ e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*B*c**2*d*
e**4*x**4/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2
*sqrt(d + e*x)) + 6*B*c**2*e**5*x**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*s
qrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**2*x + 2*A*a*c*x**3
/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x**4/2 + B*c**2*x**6/6)/d**(7/2), Tru
e))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.299633, size = 431, normalized size = 2.01 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{2} e^{24} - 25 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{2} d e^{24} + 150 \, \sqrt{x e + d} B c^{2} d^{2} e^{24} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{2} e^{25} - 60 \, \sqrt{x e + d} A c^{2} d e^{25} + 30 \, \sqrt{x e + d} B a c e^{26}\right )} e^{\left (-30\right )} + \frac{2 \,{\left (150 \,{\left (x e + d\right )}^{2} B c^{2} d^{3} - 25 \,{\left (x e + d\right )} B c^{2} d^{4} + 3 \, B c^{2} d^{5} - 90 \,{\left (x e + d\right )}^{2} A c^{2} d^{2} e + 20 \,{\left (x e + d\right )} A c^{2} d^{3} e - 3 \, A c^{2} d^{4} e + 90 \,{\left (x e + d\right )}^{2} B a c d e^{2} - 30 \,{\left (x e + d\right )} B a c d^{2} e^{2} + 6 \, B a c d^{3} e^{2} - 30 \,{\left (x e + d\right )}^{2} A a c e^{3} + 20 \,{\left (x e + d\right )} A a c d e^{3} - 6 \, A a c d^{2} e^{3} - 5 \,{\left (x e + d\right )} B a^{2} e^{4} + 3 \, B a^{2} d e^{4} - 3 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^2*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

2/15*(3*(x*e + d)^(5/2)*B*c^2*e^24 - 25*(x*e + d)^(3/2)*B*c^2*d*e^24 + 150*sqrt(
x*e + d)*B*c^2*d^2*e^24 + 5*(x*e + d)^(3/2)*A*c^2*e^25 - 60*sqrt(x*e + d)*A*c^2*
d*e^25 + 30*sqrt(x*e + d)*B*a*c*e^26)*e^(-30) + 2/15*(150*(x*e + d)^2*B*c^2*d^3
- 25*(x*e + d)*B*c^2*d^4 + 3*B*c^2*d^5 - 90*(x*e + d)^2*A*c^2*d^2*e + 20*(x*e +
d)*A*c^2*d^3*e - 3*A*c^2*d^4*e + 90*(x*e + d)^2*B*a*c*d*e^2 - 30*(x*e + d)*B*a*c
*d^2*e^2 + 6*B*a*c*d^3*e^2 - 30*(x*e + d)^2*A*a*c*e^3 + 20*(x*e + d)*A*a*c*d*e^3
- 6*A*a*c*d^2*e^3 - 5*(x*e + d)*B*a^2*e^4 + 3*B*a^2*d*e^4 - 3*A*a^2*e^5)*e^(-6)
/(x*e + d)^(5/2)