3.1726 \(\int (a+b x)^2 (A+B x) (d+e x)^{7/2} \, dx\)
Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
[Out]
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(1
5*e^4)
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Rubi [A] time = 0.0716886, antiderivative size = 128, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{77} \[ -\frac{2 b (d+e x)^{13/2} (-2 a B e-A b e+3 b B d)}{13 e^4}+\frac{2 (d+e x)^{11/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{11 e^4}-\frac{2 (d+e x)^{9/2} (b d-a e)^2 (B d-A e)}{9 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(11/2))/(11*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^4) + (2*b^2*B*(d + e*x)^(15/2))/(1
5*e^4)
Rule 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rubi steps
\begin{align*} \int (a+b x)^2 (A+B x) (d+e x)^{7/2} \, dx &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^{7/2}}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{9/2}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{11/2}}{e^3}+\frac{b^2 B (d+e x)^{13/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2 (B d-A e) (d+e x)^{9/2}}{9 e^4}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{11/2}}{11 e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{13/2}}{13 e^4}+\frac{2 b^2 B (d+e x)^{15/2}}{15 e^4}\\ \end{align*}
Mathematica [A] time = 0.140317, size = 107, normalized size = 0.84 \[ \frac{2 (d+e x)^{9/2} \left (-495 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+585 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-715 (b d-a e)^2 (B d-A e)+429 b^2 B (d+e x)^3\right )}{6435 e^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(7/2),x]
[Out]
(2*(d + e*x)^(9/2)*(-715*(b*d - a*e)^2*(B*d - A*e) + 585*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) - 4
95*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 429*b^2*B*(d + e*x)^3))/(6435*e^4)
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Maple [A] time = 0.005, size = 169, normalized size = 1.3 \begin{align*}{\frac{858\,B{b}^{2}{x}^{3}{e}^{3}+990\,A{b}^{2}{e}^{3}{x}^{2}+1980\,Bab{e}^{3}{x}^{2}-396\,B{b}^{2}d{e}^{2}{x}^{2}+2340\,Aab{e}^{3}x-360\,A{b}^{2}d{e}^{2}x+1170\,B{a}^{2}{e}^{3}x-720\,Babd{e}^{2}x+144\,B{b}^{2}{d}^{2}ex+1430\,{a}^{2}A{e}^{3}-520\,Aabd{e}^{2}+80\,A{b}^{2}{d}^{2}e-260\,B{a}^{2}d{e}^{2}+160\,Bab{d}^{2}e-32\,B{b}^{2}{d}^{3}}{6435\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x)
[Out]
2/6435*(e*x+d)^(9/2)*(429*B*b^2*e^3*x^3+495*A*b^2*e^3*x^2+990*B*a*b*e^3*x^2-198*B*b^2*d*e^2*x^2+1170*A*a*b*e^3
*x-180*A*b^2*d*e^2*x+585*B*a^2*e^3*x-360*B*a*b*d*e^2*x+72*B*b^2*d^2*e*x+715*A*a^2*e^3-260*A*a*b*d*e^2+40*A*b^2
*d^2*e-130*B*a^2*d*e^2+80*B*a*b*d^2*e-16*B*b^2*d^3)/e^4
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Maxima [A] time = 1.01771, size = 215, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (429 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{2} - 495 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 585 \,{\left (3 \, B b^{2} d^{2} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 715 \,{\left (B b^{2} d^{3} - A a^{2} e^{3} -{\left (2 \, B a b + A b^{2}\right )} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="maxima")
[Out]
2/6435*(429*(e*x + d)^(15/2)*B*b^2 - 495*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(13/2) + 585*(3*B*b^2*d^2
- 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(11/2) - 715*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b +
A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(9/2))/e^4
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Fricas [B] time = 1.51486, size = 950, normalized size = 7.42 \begin{align*} \frac{2 \,{\left (429 \, B b^{2} e^{7} x^{7} - 16 \, B b^{2} d^{7} + 715 \, A a^{2} d^{4} e^{3} + 40 \,{\left (2 \, B a b + A b^{2}\right )} d^{6} e - 130 \,{\left (B a^{2} + 2 \, A a b\right )} d^{5} e^{2} + 33 \,{\left (46 \, B b^{2} d e^{6} + 15 \,{\left (2 \, B a b + A b^{2}\right )} e^{7}\right )} x^{6} + 9 \,{\left (206 \, B b^{2} d^{2} e^{5} + 200 \,{\left (2 \, B a b + A b^{2}\right )} d e^{6} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} e^{7}\right )} x^{5} + 5 \,{\left (160 \, B b^{2} d^{3} e^{4} + 143 \, A a^{2} e^{7} + 458 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{5} + 442 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{6}\right )} x^{4} + 5 \,{\left (B b^{2} d^{4} e^{3} + 572 \, A a^{2} d e^{6} + 212 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{4} + 598 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} d^{5} e^{2} - 1430 \, A a^{2} d^{2} e^{5} - 5 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e^{3} - 520 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{4}\right )} x^{2} +{\left (8 \, B b^{2} d^{6} e + 2860 \, A a^{2} d^{3} e^{4} - 20 \,{\left (2 \, B a b + A b^{2}\right )} d^{5} e^{2} + 65 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{6435 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="fricas")
[Out]
2/6435*(429*B*b^2*e^7*x^7 - 16*B*b^2*d^7 + 715*A*a^2*d^4*e^3 + 40*(2*B*a*b + A*b^2)*d^6*e - 130*(B*a^2 + 2*A*a
*b)*d^5*e^2 + 33*(46*B*b^2*d*e^6 + 15*(2*B*a*b + A*b^2)*e^7)*x^6 + 9*(206*B*b^2*d^2*e^5 + 200*(2*B*a*b + A*b^2
)*d*e^6 + 65*(B*a^2 + 2*A*a*b)*e^7)*x^5 + 5*(160*B*b^2*d^3*e^4 + 143*A*a^2*e^7 + 458*(2*B*a*b + A*b^2)*d^2*e^5
+ 442*(B*a^2 + 2*A*a*b)*d*e^6)*x^4 + 5*(B*b^2*d^4*e^3 + 572*A*a^2*d*e^6 + 212*(2*B*a*b + A*b^2)*d^3*e^4 + 598
*(B*a^2 + 2*A*a*b)*d^2*e^5)*x^3 - 3*(2*B*b^2*d^5*e^2 - 1430*A*a^2*d^2*e^5 - 5*(2*B*a*b + A*b^2)*d^4*e^3 - 520*
(B*a^2 + 2*A*a*b)*d^3*e^4)*x^2 + (8*B*b^2*d^6*e + 2860*A*a^2*d^3*e^4 - 20*(2*B*a*b + A*b^2)*d^5*e^2 + 65*(B*a^
2 + 2*A*a*b)*d^4*e^3)*x)*sqrt(e*x + d)/e^4
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Sympy [A] time = 11.0143, size = 1020, normalized size = 7.97 \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)**2*(B*x+A)*(e*x+d)**(7/2),x)
[Out]
Piecewise((2*A*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**2*d**3*x*sqrt(d + e*x)/9 + 4*A*a**2*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**2*e**3*x**4*sqrt(d + e*x)/9 - 8*A*a*b*d**5*sqrt(d + e*
x)/(99*e**2) + 4*A*a*b*d**4*x*sqrt(d + e*x)/(99*e) + 32*A*a*b*d**3*x**2*sqrt(d + e*x)/33 + 184*A*a*b*d**2*e*x*
*3*sqrt(d + e*x)/99 + 136*A*a*b*d*e**2*x**4*sqrt(d + e*x)/99 + 4*A*a*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*b**2*
d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*A*b**2*d**4*x**2*sqrt(d + e*x)/
(429*e) + 424*A*b**2*d**3*x**3*sqrt(d + e*x)/1287 + 916*A*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*b**2*d*e*
*2*x**5*sqrt(d + e*x)/143 + 2*A*b**2*e**3*x**6*sqrt(d + e*x)/13 - 4*B*a**2*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*
a**2*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**2*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a**2*d**2*e*x**3*sqrt(d + e*x)/
99 + 68*B*a**2*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**2*e**3*x**5*sqrt(d + e*x)/11 + 32*B*a*b*d**6*sqrt(d + e*x
)/(1287*e**3) - 16*B*a*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 4*B*a*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 848*B*a*
b*d**3*x**3*sqrt(d + e*x)/1287 + 1832*B*a*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 160*B*a*b*d*e**2*x**5*sqrt(d + e*
x)/143 + 4*B*a*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*b**2*d**7*sqrt(d + e*x)/(6435*e**4) + 16*B*b**2*d**6*x*sqrt
(d + e*x)/(6435*e**3) - 4*B*b**2*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*b**2*d**4*x**3*sqrt(d + e*x)/(1287*
e) + 320*B*b**2*d**3*x**4*sqrt(d + e*x)/1287 + 412*B*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*b**2*d*e**2*x**
6*sqrt(d + e*x)/195 + 2*B*b**2*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)), (d**(7/2)*(A*a**2*x + A*a*b*x**2 + A*b**
2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4), True))
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Giac [B] time = 2.06626, size = 1854, normalized size = 14.48 \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)^2*(B*x+A)*(e*x+d)^(7/2),x, algorithm="giac")
[Out]
2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a^2*d^3*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*A*a*b*d^3*e^(-1) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a*b*
d^3*e^(-2) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*b^2*d^3*e^(-2) + 143*(
35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*b^2*d^3*e^(-
3) + 15015*(x*e + d)^(3/2)*A*a^2*d^3 + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^
2)*B*a^2*d^2*e^(-1) + 2574*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a*b*d^2*e^(-
1) + 858*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*B*a*
b*d^2*e^(-2) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)
*d^3)*A*b^2*d^2*e^(-2) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(
x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*b^2*d^2*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*
d)*A*a^2*d^2 + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)
*d^3)*B*a^2*d*e^(-1) + 858*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e +
d)^(3/2)*d^3)*A*a*b*d*e^(-1) + 78*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 -
2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a*b*d*e^(-2) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d
)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*b^2*d*e^(-2) + 1
5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 90
09*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*b^2*d*e^(-3) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5
/2)*d + 35*(x*e + d)^(3/2)*d^2)*A*a^2*d + 13*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(
7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*a^2*e^(-1) + 26*(315*(x*e + d)^(11/2) - 1540
*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*A*a*b*e^(
-1) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d
^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*B*a*b*e^(-2) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e +
d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e +
d)^(3/2)*d^5)*A*b^2*e^(-2) + (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 -
100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6
)*B*b^2*e^(-3) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/
2)*d^3)*A*a^2)*e^(-1)