3.1785 \(\int (A+B x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=128 \[ -\frac{2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac{2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac{2 b^2 B (d+e x)^{13/2}}{13 e^4} \]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(9/2))/(9*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^2*B*(d + e*x)^(13/2))/(13*
e^4)

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Rubi [A]  time = 0.0566992, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 77} \[ -\frac{2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac{2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac{2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac{2 b^2 B (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(9/2))/(9*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^2*B*(d + e*x)^(13/2))/(13*
e^4)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^{5/2}}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{7/2}}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^3}+\frac{b^2 B (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac{2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac{2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac{2 b^2 B (d+e x)^{13/2}}{13 e^4}\\ \end{align*}

Mathematica [A]  time = 0.126176, size = 107, normalized size = 0.84 \[ \frac{2 (d+e x)^{7/2} \left (-819 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+1001 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-1287 (b d-a e)^2 (B d-A e)+693 b^2 B (d+e x)^3\right )}{9009 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(-1287*(b*d - a*e)^2*(B*d - A*e) + 1001*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) -
819*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 693*b^2*B*(d + e*x)^3))/(9009*e^4)

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Maple [A]  time = 0.007, size = 169, normalized size = 1.3 \begin{align*}{\frac{1386\,{b}^{2}B{x}^{3}{e}^{3}+1638\,A{b}^{2}{e}^{3}{x}^{2}+3276\,Bab{e}^{3}{x}^{2}-756\,B{b}^{2}d{e}^{2}{x}^{2}+4004\,Axab{e}^{3}-728\,Ax{b}^{2}d{e}^{2}+2002\,Bx{a}^{2}{e}^{3}-1456\,Bxabd{e}^{2}+336\,B{b}^{2}{d}^{2}ex+2574\,A{a}^{2}{e}^{3}-1144\,Aabd{e}^{2}+208\,A{b}^{2}{d}^{2}e-572\,B{a}^{2}d{e}^{2}+416\,Bab{d}^{2}e-96\,{b}^{2}B{d}^{3}}{9009\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/9009*(e*x+d)^(7/2)*(693*B*b^2*e^3*x^3+819*A*b^2*e^3*x^2+1638*B*a*b*e^3*x^2-378*B*b^2*d*e^2*x^2+2002*A*a*b*e^
3*x-364*A*b^2*d*e^2*x+1001*B*a^2*e^3*x-728*B*a*b*d*e^2*x+168*B*b^2*d^2*e*x+1287*A*a^2*e^3-572*A*a*b*d*e^2+104*
A*b^2*d^2*e-286*B*a^2*d*e^2+208*B*a*b*d^2*e-48*B*b^2*d^3)/e^4

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Maxima [A]  time = 0.974161, size = 215, normalized size = 1.68 \begin{align*} \frac{2 \,{\left (693 \,{\left (e x + d\right )}^{\frac{13}{2}} B b^{2} - 819 \,{\left (3 \, B b^{2} d -{\left (2 \, B a b + A b^{2}\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\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{9}{2}} - 1287 \,{\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{7}{2}}\right )}}{9009 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/9009*(693*(e*x + d)^(13/2)*B*b^2 - 819*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(11/2) + 1001*(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)^(9/2) - 1287*(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)^(7/2))/e^4

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Fricas [B]  time = 1.22601, size = 813, normalized size = 6.35 \begin{align*} \frac{2 \,{\left (693 \, B b^{2} e^{6} x^{6} - 48 \, B b^{2} d^{6} + 1287 \, A a^{2} d^{3} e^{3} + 104 \,{\left (2 \, B a b + A b^{2}\right )} d^{5} e - 286 \,{\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{2} + 63 \,{\left (27 \, B b^{2} d e^{5} + 13 \,{\left (2 \, B a b + A b^{2}\right )} e^{6}\right )} x^{5} + 7 \,{\left (159 \, B b^{2} d^{2} e^{4} + 299 \,{\left (2 \, B a b + A b^{2}\right )} d e^{5} + 143 \,{\left (B a^{2} + 2 \, A a b\right )} e^{6}\right )} x^{4} +{\left (15 \, B b^{2} d^{3} e^{3} + 1287 \, A a^{2} e^{6} + 1469 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{4} + 2717 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{5}\right )} x^{3} - 3 \,{\left (6 \, B b^{2} d^{4} e^{2} - 1287 \, A a^{2} d e^{5} - 13 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e^{3} - 715 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{4}\right )} x^{2} +{\left (24 \, B b^{2} d^{5} e + 3861 \, A a^{2} d^{2} e^{4} - 52 \,{\left (2 \, B a b + A b^{2}\right )} d^{4} e^{2} + 143 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{3}\right )} x\right )} \sqrt{e x + d}}{9009 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^2*e^6*x^6 - 48*B*b^2*d^6 + 1287*A*a^2*d^3*e^3 + 104*(2*B*a*b + A*b^2)*d^5*e - 286*(B*a^2 + 2*A
*a*b)*d^4*e^2 + 63*(27*B*b^2*d*e^5 + 13*(2*B*a*b + A*b^2)*e^6)*x^5 + 7*(159*B*b^2*d^2*e^4 + 299*(2*B*a*b + A*b
^2)*d*e^5 + 143*(B*a^2 + 2*A*a*b)*e^6)*x^4 + (15*B*b^2*d^3*e^3 + 1287*A*a^2*e^6 + 1469*(2*B*a*b + A*b^2)*d^2*e
^4 + 2717*(B*a^2 + 2*A*a*b)*d*e^5)*x^3 - 3*(6*B*b^2*d^4*e^2 - 1287*A*a^2*d*e^5 - 13*(2*B*a*b + A*b^2)*d^3*e^3
- 715*(B*a^2 + 2*A*a*b)*d^2*e^4)*x^2 + (24*B*b^2*d^5*e + 3861*A*a^2*d^2*e^4 - 52*(2*B*a*b + A*b^2)*d^4*e^2 + 1
43*(B*a^2 + 2*A*a*b)*d^3*e^3)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 5.59354, size = 857, normalized size = 6.7 \begin{align*} \begin{cases} \frac{2 A a^{2} d^{3} \sqrt{d + e x}}{7 e} + \frac{6 A a^{2} d^{2} x \sqrt{d + e x}}{7} + \frac{6 A a^{2} d e x^{2} \sqrt{d + e x}}{7} + \frac{2 A a^{2} e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{8 A a b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{4 A a b d^{3} x \sqrt{d + e x}}{63 e} + \frac{20 A a b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{76 A a b d e x^{3} \sqrt{d + e x}}{63} + \frac{4 A a b e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 A b^{2} d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 A b^{2} d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 A b^{2} d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 A b^{2} d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 A b^{2} d e x^{4} \sqrt{d + e x}}{99} + \frac{2 A b^{2} e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{4 B a^{2} d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 B a^{2} d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 B a^{2} d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 B a^{2} d e x^{3} \sqrt{d + e x}}{63} + \frac{2 B a^{2} e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{32 B a b d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{16 B a b d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{4 B a b d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{452 B a b d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{92 B a b d e x^{4} \sqrt{d + e x}}{99} + \frac{4 B a b e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{32 B b^{2} d^{6} \sqrt{d + e x}}{3003 e^{4}} + \frac{16 B b^{2} d^{5} x \sqrt{d + e x}}{3003 e^{3}} - \frac{4 B b^{2} d^{4} x^{2} \sqrt{d + e x}}{1001 e^{2}} + \frac{10 B b^{2} d^{3} x^{3} \sqrt{d + e x}}{3003 e} + \frac{106 B b^{2} d^{2} x^{4} \sqrt{d + e x}}{429} + \frac{54 B b^{2} d e x^{5} \sqrt{d + e x}}{143} + \frac{2 B b^{2} e^{2} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (A a^{2} x + A a b x^{2} + \frac{A b^{2} x^{3}}{3} + \frac{B a^{2} x^{2}}{2} + \frac{2 B a b x^{3}}{3} + \frac{B b^{2} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*A*a**2*d**3*sqrt(d + e*x)/(7*e) + 6*A*a**2*d**2*x*sqrt(d + e*x)/7 + 6*A*a**2*d*e*x**2*sqrt(d + e*
x)/7 + 2*A*a**2*e**2*x**3*sqrt(d + e*x)/7 - 8*A*a*b*d**4*sqrt(d + e*x)/(63*e**2) + 4*A*a*b*d**3*x*sqrt(d + e*x
)/(63*e) + 20*A*a*b*d**2*x**2*sqrt(d + e*x)/21 + 76*A*a*b*d*e*x**3*sqrt(d + e*x)/63 + 4*A*a*b*e**2*x**4*sqrt(d
+ e*x)/9 + 16*A*b**2*d**5*sqrt(d + e*x)/(693*e**3) - 8*A*b**2*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*A*b**2*d**3
*x**2*sqrt(d + e*x)/(231*e) + 226*A*b**2*d**2*x**3*sqrt(d + e*x)/693 + 46*A*b**2*d*e*x**4*sqrt(d + e*x)/99 + 2
*A*b**2*e**2*x**5*sqrt(d + e*x)/11 - 4*B*a**2*d**4*sqrt(d + e*x)/(63*e**2) + 2*B*a**2*d**3*x*sqrt(d + e*x)/(63
*e) + 10*B*a**2*d**2*x**2*sqrt(d + e*x)/21 + 38*B*a**2*d*e*x**3*sqrt(d + e*x)/63 + 2*B*a**2*e**2*x**4*sqrt(d +
e*x)/9 + 32*B*a*b*d**5*sqrt(d + e*x)/(693*e**3) - 16*B*a*b*d**4*x*sqrt(d + e*x)/(693*e**2) + 4*B*a*b*d**3*x**
2*sqrt(d + e*x)/(231*e) + 452*B*a*b*d**2*x**3*sqrt(d + e*x)/693 + 92*B*a*b*d*e*x**4*sqrt(d + e*x)/99 + 4*B*a*b
*e**2*x**5*sqrt(d + e*x)/11 - 32*B*b**2*d**6*sqrt(d + e*x)/(3003*e**4) + 16*B*b**2*d**5*x*sqrt(d + e*x)/(3003*
e**3) - 4*B*b**2*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*B*b**2*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*B*b**2
*d**2*x**4*sqrt(d + e*x)/429 + 54*B*b**2*d*e*x**5*sqrt(d + e*x)/143 + 2*B*b**2*e**2*x**6*sqrt(d + e*x)/13, Ne(
e, 0)), (d**(5/2)*(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), Tr
ue))

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Giac [B]  time = 1.21071, size = 1224, normalized size = 9.56 \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)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^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^2*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e +
d)^(3/2)*d)*A*a*b*d^2*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^2*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^2*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^2*e^(-
3) + 15015*(x*e + d)^(3/2)*A*a^2*d^2 + 858*(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*e^(-1) + 1716*(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*e^(-1) +
572*(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*e
^(-2) + 286*(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*e^(-2) + 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)*B*b^2*d*e^(-3) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*a^2*d
+ 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*a^2*e
^(-1) + 286*(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*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)*B*a*b*e^(-2) + 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)*A*b^2*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)*B*b^2*e^(-3) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2
)*d^2)*A*a^2)*e^(-1)