3.1214 \(\int (A+B x) (d+e x)^{5/2} (b x+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 (d+e x)^{11/2} (-A c e-b B e+3 B c d)}{11 e^4}+\frac{2 (d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{9 e^4}-\frac{2 d (d+e x)^{7/2} (B d-A e) (c d-b e)}{7 e^4}+\frac{2 B c (d+e x)^{13/2}}{13 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0787755, antiderivative size = 126, 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 = {771} \[ -\frac{2 (d+e x)^{11/2} (-A c e-b B e+3 B c d)}{11 e^4}+\frac{2 (d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{9 e^4}-\frac{2 d (d+e x)^{7/2} (B d-A e) (c d-b e)}{7 e^4}+\frac{2 B c (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.129849, size = 113, normalized size = 0.9 \[ \frac{2 (d+e x)^{7/2} \left (13 A e \left (11 b e (7 e x-2 d)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+B \left (13 b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+c \left (168 d^2 e x-48 d^3-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(13*A*e*(11*b*e*(-2*d + 7*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) + B*(13*b*e*(8*d^2 - 28
*d*e*x + 63*e^2*x^2) + c*(-48*d^3 + 168*d^2*e*x - 378*d*e^2*x^2 + 693*e^3*x^3))))/(9009*e^4)

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Maple [A]  time = 0.005, size = 121, normalized size = 1. \begin{align*} -{\frac{-1386\,Bc{x}^{3}{e}^{3}-1638\,Ac{e}^{3}{x}^{2}-1638\,Bb{e}^{3}{x}^{2}+756\,Bcd{e}^{2}{x}^{2}-2002\,Ab{e}^{3}x+728\,Acd{e}^{2}x+728\,Bbd{e}^{2}x-336\,Bc{d}^{2}ex+572\,Abd{e}^{2}-208\,Ac{d}^{2}e-208\,Bb{d}^{2}e+96\,Bc{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)*(c*x^2+b*x),x)

[Out]

-2/9009*(e*x+d)^(7/2)*(-693*B*c*e^3*x^3-819*A*c*e^3*x^2-819*B*b*e^3*x^2+378*B*c*d*e^2*x^2-1001*A*b*e^3*x+364*A
*c*d*e^2*x+364*B*b*d*e^2*x-168*B*c*d^2*e*x+286*A*b*d*e^2-104*A*c*d^2*e-104*B*b*d^2*e+48*B*c*d^3)/e^4

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Maxima [A]  time = 1.09752, size = 151, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (693 \,{\left (e x + d\right )}^{\frac{13}{2}} B c - 819 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 1287 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\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)*(c*x^2+b*x),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.7812, size = 545, normalized size = 4.33 \begin{align*} \frac{2 \,{\left (693 \, B c e^{6} x^{6} - 48 \, B c d^{6} - 286 \, A b d^{4} e^{2} + 104 \,{\left (B b + A c\right )} d^{5} e + 63 \,{\left (27 \, B c d e^{5} + 13 \,{\left (B b + A c\right )} e^{6}\right )} x^{5} + 7 \,{\left (159 \, B c d^{2} e^{4} + 143 \, A b e^{6} + 299 \,{\left (B b + A c\right )} d e^{5}\right )} x^{4} +{\left (15 \, B c d^{3} e^{3} + 2717 \, A b d e^{5} + 1469 \,{\left (B b + A c\right )} d^{2} e^{4}\right )} x^{3} - 3 \,{\left (6 \, B c d^{4} e^{2} - 715 \, A b d^{2} e^{4} - 13 \,{\left (B b + A c\right )} d^{3} e^{3}\right )} x^{2} +{\left (24 \, B c d^{5} e + 143 \, A b d^{3} e^{3} - 52 \,{\left (B b + A c\right )} d^{4} e^{2}\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)*(c*x^2+b*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.25675, size = 581, normalized size = 4.61 \begin{align*} \begin{cases} - \frac{4 A b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{2 A b d^{3} x \sqrt{d + e x}}{63 e} + \frac{10 A b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{38 A b d e x^{3} \sqrt{d + e x}}{63} + \frac{2 A b e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 A c d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 A c d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 A c d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 A c d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 A c d e x^{4} \sqrt{d + e x}}{99} + \frac{2 A c e^{2} x^{5} \sqrt{d + e x}}{11} + \frac{16 B b d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 B b d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 B b d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 B b d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 B b d e x^{4} \sqrt{d + e x}}{99} + \frac{2 B b e^{2} x^{5} \sqrt{d + e x}}{11} - \frac{32 B c d^{6} \sqrt{d + e x}}{3003 e^{4}} + \frac{16 B c d^{5} x \sqrt{d + e x}}{3003 e^{3}} - \frac{4 B c d^{4} x^{2} \sqrt{d + e x}}{1001 e^{2}} + \frac{10 B c d^{3} x^{3} \sqrt{d + e x}}{3003 e} + \frac{106 B c d^{2} x^{4} \sqrt{d + e x}}{429} + \frac{54 B c d e x^{5} \sqrt{d + e x}}{143} + \frac{2 B c e^{2} x^{6} \sqrt{d + e x}}{13} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (\frac{A b x^{2}}{2} + \frac{A c x^{3}}{3} + \frac{B b x^{3}}{3} + \frac{B c 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)*(c*x**2+b*x),x)

[Out]

Piecewise((-4*A*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*A*b*d**3*x*sqrt(d + e*x)/(63*e) + 10*A*b*d**2*x**2*sqrt(d +
e*x)/21 + 38*A*b*d*e*x**3*sqrt(d + e*x)/63 + 2*A*b*e**2*x**4*sqrt(d + e*x)/9 + 16*A*c*d**5*sqrt(d + e*x)/(693
*e**3) - 8*A*c*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*A*c*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*A*c*d**2*x**3*sqr
t(d + e*x)/693 + 46*A*c*d*e*x**4*sqrt(d + e*x)/99 + 2*A*c*e**2*x**5*sqrt(d + e*x)/11 + 16*B*b*d**5*sqrt(d + e*
x)/(693*e**3) - 8*B*b*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*B*b*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*B*b*d**2*x
**3*sqrt(d + e*x)/693 + 46*B*b*d*e*x**4*sqrt(d + e*x)/99 + 2*B*b*e**2*x**5*sqrt(d + e*x)/11 - 32*B*c*d**6*sqrt
(d + e*x)/(3003*e**4) + 16*B*c*d**5*x*sqrt(d + e*x)/(3003*e**3) - 4*B*c*d**4*x**2*sqrt(d + e*x)/(1001*e**2) +
10*B*c*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*B*c*d**2*x**4*sqrt(d + e*x)/429 + 54*B*c*d*e*x**5*sqrt(d + e*x)/
143 + 2*B*c*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(5/2)*(A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/
4), True))

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Giac [B]  time = 1.36446, size = 903, normalized size = 7.17 \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)*(c*x^2+b*x),x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*A*b*d^2*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*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*c*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 - 1
05*(x*e + d)^(3/2)*d^3)*B*c*d^2*e^(-3) + 858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d
^2)*A*b*d*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)*B*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*c*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*c*d*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*b*e^(-1) + 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*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*c*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*c*e^(-3))*e^(-1)