3.1718 \(\int (a+b x) (A+B x) (d+e x)^{7/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac {2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac {2 b B (d+e x)^{13/2}}{13 e^3} \]

[Out]

2/9*(-a*e+b*d)*(-A*e+B*d)*(e*x+d)^(9/2)/e^3-2/11*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(11/2)/e^3+2/13*b*B*(e*x+d)^(1
3/2)/e^3

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {2 (d+e x)^{11/2} (-a B e-A b e+2 b B d)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e) (B d-A e)}{9 e^3}+\frac {2 b B (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.11, size = 70, normalized size = 0.84 \[ \frac {2 (d+e x)^{9/2} \left (13 a e (11 A e-2 B d+9 B e x)+13 A b e (9 e x-2 d)+b B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(9/2)*(13*A*b*e*(-2*d + 9*e*x) + 13*a*e*(-2*B*d + 11*A*e + 9*B*e*x) + b*B*(8*d^2 - 36*d*e*x + 99*
e^2*x^2)))/(1287*e^3)

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fricas [B]  time = 1.07, size = 230, normalized size = 2.77 \[ \frac {2 \, {\left (99 \, B b e^{6} x^{6} + 8 \, B b d^{6} + 143 \, A a d^{4} e^{2} - 26 \, {\left (B a + A b\right )} d^{5} e + 9 \, {\left (40 \, B b d e^{5} + 13 \, {\left (B a + A b\right )} e^{6}\right )} x^{5} + {\left (458 \, B b d^{2} e^{4} + 143 \, A a e^{6} + 442 \, {\left (B a + A b\right )} d e^{5}\right )} x^{4} + 2 \, {\left (106 \, B b d^{3} e^{3} + 286 \, A a d e^{5} + 299 \, {\left (B a + A b\right )} d^{2} e^{4}\right )} x^{3} + 3 \, {\left (B b d^{4} e^{2} + 286 \, A a d^{2} e^{4} + 104 \, {\left (B a + A b\right )} d^{3} e^{3}\right )} x^{2} - {\left (4 \, B b d^{5} e - 572 \, A a d^{3} e^{3} - 13 \, {\left (B a + A b\right )} d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/1287*(99*B*b*e^6*x^6 + 8*B*b*d^6 + 143*A*a*d^4*e^2 - 26*(B*a + A*b)*d^5*e + 9*(40*B*b*d*e^5 + 13*(B*a + A*b)
*e^6)*x^5 + (458*B*b*d^2*e^4 + 143*A*a*e^6 + 442*(B*a + A*b)*d*e^5)*x^4 + 2*(106*B*b*d^3*e^3 + 286*A*a*d*e^5 +
299*(B*a + A*b)*d^2*e^4)*x^3 + 3*(B*b*d^4*e^2 + 286*A*a*d^2*e^4 + 104*(B*a + A*b)*d^3*e^3)*x^2 - (4*B*b*d^5*e
- 572*A*a*d^3*e^3 - 13*(B*a + A*b)*d^4*e^2)*x)*sqrt(e*x + d)/e^3

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giac [B]  time = 1.37, size = 1108, normalized size = 13.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*d^4*e^(-1) + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)
*d)*A*b*d^4*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^4*e^(-2) + 1
2012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*d^3*e^(-1) + 12012*(3*(x*e + d)^(5/
2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b*d^3*e^(-1) + 5148*(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^3*e^(-2) + 45045*sqrt(x*e + d)*A*a*d^4 + 60060*((x*
e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a*d^3 + 7722*(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*a*d^2*e^(-1) + 7722*(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^2*e^(-1) + 858*(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^2*e^(-2) + 18018*(3*(x*e + d)^(5/2) - 1
0*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*d^2 + 572*(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*a*d*e^(-1) + 572*(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*d*e^(-1
) + 260*(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 + 11
55*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*b*d*e^(-2) + 5148*(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*a*d + 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)*B*a*e^(-1)
+ 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*b*e^(-1) + 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 - 6006*(x*e + d)^(3/2)*d^5 +
3003*sqrt(x*e + d)*d^6)*B*b*e^(-2) + 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*a)*e^(-1)

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maple [A]  time = 0.00, size = 73, normalized size = 0.88 \[ \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 B b \,x^{2} e^{2}+117 A b \,e^{2} x +117 B a \,e^{2} x -36 B b d e x +143 A a \,e^{2}-26 A b d e -26 B a d e +8 B b \,d^{2}\right )}{1287 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x)

[Out]

2/1287*(e*x+d)^(9/2)*(99*B*b*e^2*x^2+117*A*b*e^2*x+117*B*a*e^2*x-36*B*b*d*e*x+143*A*a*e^2-26*A*b*d*e-26*B*a*d*
e+8*B*b*d^2)/e^3

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maxima [A]  time = 0.56, size = 75, normalized size = 0.90 \[ \frac {2 \, {\left (99 \, {\left (e x + d\right )}^{\frac {13}{2}} B b - 117 \, {\left (2 \, B b d - {\left (B a + A b\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 143 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{1287 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*B*b - 117*(2*B*b*d - (B*a + A*b)*e)*(e*x + d)^(11/2) + 143*(B*b*d^2 + A*a*e^2 - (B
*a + A*b)*d*e)*(e*x + d)^(9/2))/e^3

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mupad [B]  time = 0.09, size = 80, normalized size = 0.96 \[ \frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,B\,b\,{\left (d+e\,x\right )}^2+143\,A\,a\,e^2+143\,B\,b\,d^2+117\,A\,b\,e\,\left (d+e\,x\right )+117\,B\,a\,e\,\left (d+e\,x\right )-234\,B\,b\,d\,\left (d+e\,x\right )-143\,A\,b\,d\,e-143\,B\,a\,d\,e\right )}{1287\,e^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)*(d + e*x)^(7/2),x)

[Out]

(2*(d + e*x)^(9/2)*(99*B*b*(d + e*x)^2 + 143*A*a*e^2 + 143*B*b*d^2 + 117*A*b*e*(d + e*x) + 117*B*a*e*(d + e*x)
- 234*B*b*d*(d + e*x) - 143*A*b*d*e - 143*B*a*d*e))/(1287*e^3)

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sympy [A]  time = 7.96, size = 578, normalized size = 6.96 \[ \begin {cases} \frac {2 A a d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {4 A b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 A b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 A b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 A b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 A b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 A b e^{3} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 B b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 B b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 B b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 B b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 B b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 B b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 B b e^{3} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a x + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)*(e*x+d)**(7/2),x)

[Out]

Piecewise((2*A*a*d**4*sqrt(d + e*x)/(9*e) + 8*A*a*d**3*x*sqrt(d + e*x)/9 + 4*A*a*d**2*e*x**2*sqrt(d + e*x)/3 +
8*A*a*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a*e**3*x**4*sqrt(d + e*x)/9 - 4*A*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*
A*b*d**4*x*sqrt(d + e*x)/(99*e) + 16*A*b*d**3*x**2*sqrt(d + e*x)/33 + 92*A*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68
*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d + e*x)/11 - 4*B*a*d**5*sqrt(d + e*x)/(99*e**2) + 2*
B*a*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a*d**3*x**2*sqrt(d + e*x)/33 + 92*B*a*d**2*e*x**3*sqrt(d + e*x)/99 + 68
*B*a*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a*e**3*x**5*sqrt(d + e*x)/11 + 16*B*b*d**6*sqrt(d + e*x)/(1287*e**3) -
8*B*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*B*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*B*b*d**3*x**3*sqrt(d + e
*x)/1287 + 916*B*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*B*b*d*e**2*x**5*sqrt(d + e*x)/143 + 2*B*b*e**3*x**6*sqr
t(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/3), True))

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