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

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

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Rubi [A]  time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.12, size = 113, normalized size = 0.90 \begin {gather*} \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 (-48 d^3+168 d^2 e x-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.09, size = 141, normalized size = 1.12 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (1001 A b e^2 (d+e x)-1287 A b d e^2+1287 A c d^2 e-2002 A c d e (d+e x)+819 A c e (d+e x)^2+1287 b B d^2 e-2002 b B d e (d+e x)+819 b B e (d+e x)^2-1287 B c d^3+3003 B c d^2 (d+e x)-2457 B c d (d+e x)^2+693 B c (d+e x)^3\right )}{9009 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(-1287*B*c*d^3 + 1287*b*B*d^2*e + 1287*A*c*d^2*e - 1287*A*b*d*e^2 + 3003*B*c*d^2*(d + e*x)
- 2002*b*B*d*e*(d + e*x) - 2002*A*c*d*e*(d + e*x) + 1001*A*b*e^2*(d + e*x) - 2457*B*c*d*(d + e*x)^2 + 819*b*B*
e*(d + e*x)^2 + 819*A*c*e*(d + e*x)^2 + 693*B*c*(d + e*x)^3))/(9009*e^4)

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fricas [B]  time = 0.40, size = 230, normalized size = 1.83 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.21, size = 999, normalized size = 7.93

result too large to display

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*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*d^3*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*B*b*d^3*e^(-2) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*A*c*d^3*e^(-2) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*
e + d)*d^3)*B*c*d^3*e^(-3) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b*d^2*e^
(-1) + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b*d^2
*e^(-2) + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*c*
d^2*e^(-2) + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d
^3 + 315*sqrt(x*e + d)*d^4)*B*c*d^2*e^(-3) + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/
2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b*d*e^(-1) + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^
(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b*d*e^(-2) + 429*(35*(x*e + d)^(9/2) - 180*(x*e
+ d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*c*d*e^(-2) + 195*
(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e
+ d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*c*d*e^(-3) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(
x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b*e^(-1) + 65*(63*(x*e + d)^(11/2) - 3
85*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqr
t(x*e + d)*d^5)*B*b*e^(-2) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*
(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*c*e^(-2) + 15*(231*(x*e + d)^(13/2)
- 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6
006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*c*e^(-3))*e^(-1)

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maple [A]  time = 0.05, size = 121, normalized size = 0.96 \begin {gather*} -\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-693 B c \,x^{3} e^{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}\right )}{9009 e^{4}} \end {gather*}

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 = 0.46, size = 112, normalized size = 0.89 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.07, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{9\,e^4}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{11\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(9/2)*(2*A*b*e^2 + 6*B*c*d^2 - 4*A*c*d*e - 4*B*b*d*e))/(9*e^4) + ((d + e*x)^(11/2)*(2*A*c*e + 2*B*b
*e - 6*B*c*d))/(11*e^4) + (2*B*c*(d + e*x)^(13/2))/(13*e^4) - (2*d*(A*e - B*d)*(b*e - c*d)*(d + e*x)^(7/2))/(7
*e^4)

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sympy [A]  time = 4.64, size = 581, normalized size = 4.61 \begin {gather*} \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 {gather*}

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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