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

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

[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 = 0.0916629, 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, Rules used = {772} \[ \frac{4 c \sqrt{d+e x} \left (a B e^2-2 A c d e+5 B c d^2\right )}{e^6}+\frac{4 c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{e^6 \sqrt{d+e x}}-\frac{2 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{2 \left (a e^2+c d^2\right )^2 (B d-A e)}{5 e^6 (d+e x)^{5/2}}-\frac{2 c^2 (d+e x)^{3/2} (5 B d-A e)}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6} \]

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)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^2}{(d+e x)^{7/2}} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^2}{e^5 (d+e x)^{7/2}}+\frac{\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{e^5 (d+e x)^{5/2}}+\frac{2 c \left (-5 B c d^3+3 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^5 (d+e x)^{3/2}}-\frac{2 c \left (-5 B c d^2+2 A c d e-a B e^2\right )}{e^5 \sqrt{d+e x}}+\frac{c^2 (-5 B d+A e) \sqrt{d+e x}}{e^5}+\frac{B c^2 (d+e x)^{3/2}}{e^5}\right ) \, dx\\ &=\frac{2 (B d-A e) \left (c d^2+a e^2\right )^2}{5 e^6 (d+e x)^{5/2}}-\frac{2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{3 e^6 (d+e x)^{3/2}}+\frac{4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^6 \sqrt{d+e x}}+\frac{4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) \sqrt{d+e x}}{e^6}-\frac{2 c^2 (5 B d-A e) (d+e x)^{3/2}}{3 e^6}+\frac{2 B c^2 (d+e x)^{5/2}}{5 e^6}\\ \end{align*}

Mathematica [A]  time = 0.152679, size = 212, normalized size = 0.99 \[ -\frac{2 \left (3 a^2 A e^5+a^2 B e^4 (2 d+5 e x)+2 a A c e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a B c e^2 \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+A c^2 e \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )-B c^2 \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )}{15 e^6 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*a^2*A*e^5 + a^2*B*e^4*(2*d + 5*e*x) + 2*a*A*c*e^3*(8*d^2 + 20*d*e*x + 15*e^2*x^2) - 6*a*B*c*e^2*(16*d^3
+ 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + A*c^2*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 -
5*e^4*x^4) - B*c^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5)))/(15
*e^6*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 259, normalized size = 1.2 \begin{align*} -{\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\,B{a}^{2}d{e}^{4}-192\,aBc{d}^{3}{e}^{2}-512\,B{c}^{2}{d}^{5}}{15\,{e}^{6}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

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^3-8
0*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+1
28*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 = 1.00067, size = 344, normalized size = 1.61 \begin{align*} \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} \end{align*}

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, 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 = 1.49272, size = 601, normalized size = 2.81 \begin{align*} \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 )} \sqrt{e x + d}}{15 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \end{align*}

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, 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 + 3
0*(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)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e
^6)

________________________________________________________________________________________

Sympy [A]  time = 4.50323, size = 1426, normalized size = 6.66 \begin{align*} \text{result too large to display} \end{align*}

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*sqrt(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*sqrt(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*sqrt(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), True))

________________________________________________________________________________________

Giac [A]  time = 1.23115, size = 431, normalized size = 2.01 \begin{align*} \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}}} \end{align*}

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, 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)