3.1717 \(\int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac{6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac{2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac{2 b^3 B (d+e x)^{17/2}}{17 e^5} \]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*
b*e - a*B*e)*(d + e*x)^(13/2))/(13*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d
+ e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*e^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.274321, antiderivative size = 173, 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 \[ -\frac{2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac{6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac{2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac{2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac{2 b^3 B (d+e x)^{17/2}}{17 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*
d - 3*A*b*e - a*B*e)*(d + e*x)^(11/2))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*
b*e - a*B*e)*(d + e*x)^(13/2))/(13*e^5) - (2*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d
+ e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*e^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 48.3009, size = 170, normalized size = 0.98 \[ \frac{2 B b^{3} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{5}} + \frac{2 b^{2} \left (d + e x\right )^{\frac{15}{2}} \left (A b e + 3 B a e - 4 B b d\right )}{15 e^{5}} + \frac{6 b \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{13 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{11 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{9}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{3}}{9 e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(7/2),x)

[Out]

2*B*b**3*(d + e*x)**(17/2)/(17*e**5) + 2*b**2*(d + e*x)**(15/2)*(A*b*e + 3*B*a*e
- 4*B*b*d)/(15*e**5) + 6*b*(d + e*x)**(13/2)*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b
*d)/(13*e**5) + 2*(d + e*x)**(11/2)*(a*e - b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(
11*e**5) + 2*(d + e*x)**(9/2)*(A*e - B*d)*(a*e - b*d)**3/(9*e**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.373722, size = 227, normalized size = 1.31 \[ \frac{2 (d+e x)^{9/2} \left (1105 a^3 e^3 (11 A e-2 B d+9 B e x)+255 a^2 b e^2 \left (13 A e (9 e x-2 d)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-51 a b^2 e \left (B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+b^3 \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )\right )}{109395 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(d + e*x)^(9/2)*(1105*a^3*e^3*(-2*B*d + 11*A*e + 9*B*e*x) + 255*a^2*b*e^2*(13
*A*e*(-2*d + 9*e*x) + B*(8*d^2 - 36*d*e*x + 99*e^2*x^2)) - 51*a*b^2*e*(-5*A*e*(8
*d^2 - 36*d*e*x + 99*e^2*x^2) + B*(16*d^3 - 72*d^2*e*x + 198*d*e^2*x^2 - 429*e^3
*x^3)) + b^3*(17*A*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + B*(1
28*d^4 - 576*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4))))/(109
395*e^5)

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 301, normalized size = 1.7 \[{\frac{12870\,B{b}^{3}{x}^{4}{e}^{4}+14586\,A{b}^{3}{e}^{4}{x}^{3}+43758\,Ba{b}^{2}{e}^{4}{x}^{3}-6864\,B{b}^{3}d{e}^{3}{x}^{3}+50490\,Aa{b}^{2}{e}^{4}{x}^{2}-6732\,A{b}^{3}d{e}^{3}{x}^{2}+50490\,B{a}^{2}b{e}^{4}{x}^{2}-20196\,Ba{b}^{2}d{e}^{3}{x}^{2}+3168\,B{b}^{3}{d}^{2}{e}^{2}{x}^{2}+59670\,A{a}^{2}b{e}^{4}x-18360\,Aa{b}^{2}d{e}^{3}x+2448\,A{b}^{3}{d}^{2}{e}^{2}x+19890\,B{a}^{3}{e}^{4}x-18360\,B{a}^{2}bd{e}^{3}x+7344\,Ba{b}^{2}{d}^{2}{e}^{2}x-1152\,B{b}^{3}{d}^{3}ex+24310\,{a}^{3}A{e}^{4}-13260\,A{a}^{2}bd{e}^{3}+4080\,Aa{b}^{2}{d}^{2}{e}^{2}-544\,A{b}^{3}{d}^{3}e-4420\,B{a}^{3}d{e}^{3}+4080\,B{a}^{2}b{d}^{2}{e}^{2}-1632\,Ba{b}^{2}{d}^{3}e+256\,B{b}^{3}{d}^{4}}{109395\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)*(e*x+d)^(7/2),x)

[Out]

2/109395*(e*x+d)^(9/2)*(6435*B*b^3*e^4*x^4+7293*A*b^3*e^4*x^3+21879*B*a*b^2*e^4*
x^3-3432*B*b^3*d*e^3*x^3+25245*A*a*b^2*e^4*x^2-3366*A*b^3*d*e^3*x^2+25245*B*a^2*
b*e^4*x^2-10098*B*a*b^2*d*e^3*x^2+1584*B*b^3*d^2*e^2*x^2+29835*A*a^2*b*e^4*x-918
0*A*a*b^2*d*e^3*x+1224*A*b^3*d^2*e^2*x+9945*B*a^3*e^4*x-9180*B*a^2*b*d*e^3*x+367
2*B*a*b^2*d^2*e^2*x-576*B*b^3*d^3*e*x+12155*A*a^3*e^4-6630*A*a^2*b*d*e^3+2040*A*
a*b^2*d^2*e^2-272*A*b^3*d^3*e-2210*B*a^3*d*e^3+2040*B*a^2*b*d^2*e^2-816*B*a*b^2*
d^3*e+128*B*b^3*d^4)/e^5

_______________________________________________________________________________________

Maxima [A]  time = 1.37087, size = 358, normalized size = 2.07 \[ \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} B b^{3} - 7293 \,{\left (4 \, B b^{3} d -{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 25245 \,{\left (2 \, B b^{3} d^{2} -{\left (3 \, B a b^{2} + A b^{3}\right )} d e +{\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 9945 \,{\left (4 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 12155 \,{\left (B b^{3} d^{4} + A a^{3} e^{4} -{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}}\right )}}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

2/109395*(6435*(e*x + d)^(17/2)*B*b^3 - 7293*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)
*(e*x + d)^(15/2) + 25245*(2*B*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*
a*b^2)*e^2)*(e*x + d)^(13/2) - 9945*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*b^3)*d^2*e +
6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d)^(11/2) + 12155
*(B*b^3*d^4 + A*a^3*e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*
e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^(9/2))/e^5

_______________________________________________________________________________________

Fricas [A]  time = 0.226842, size = 855, normalized size = 4.94 \[ \frac{2 \,{\left (6435 \, B b^{3} e^{8} x^{8} + 128 \, B b^{3} d^{8} + 12155 \, A a^{3} d^{4} e^{4} - 272 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{7} e + 2040 \,{\left (B a^{2} b + A a b^{2}\right )} d^{6} e^{2} - 2210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5} e^{3} + 429 \,{\left (52 \, B b^{3} d e^{7} + 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{8}\right )} x^{7} + 33 \,{\left (802 \, B b^{3} d^{2} e^{6} + 782 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{7} + 765 \,{\left (B a^{2} b + A a b^{2}\right )} e^{8}\right )} x^{6} + 9 \,{\left (1212 \, B b^{3} d^{3} e^{5} + 3502 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{6} + 10200 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{7} + 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{8}\right )} x^{5} + 5 \,{\left (7 \, B b^{3} d^{4} e^{4} + 2431 \, A a^{3} e^{8} + 2720 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{5} + 23358 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{6} + 7514 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{7}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} d^{5} e^{3} - 9724 \, A a^{3} d e^{7} - 17 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{4} - 10812 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{5} - 10166 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{6}\right )} x^{3} + 3 \,{\left (16 \, B b^{3} d^{6} e^{2} + 24310 \, A a^{3} d^{2} e^{6} - 34 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e^{3} + 255 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{4} + 8840 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{5}\right )} x^{2} -{\left (64 \, B b^{3} d^{7} e - 48620 \, A a^{3} d^{3} e^{5} - 136 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} e^{2} + 1020 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} e^{3} - 1105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e^{4}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

2/109395*(6435*B*b^3*e^8*x^8 + 128*B*b^3*d^8 + 12155*A*a^3*d^4*e^4 - 272*(3*B*a*
b^2 + A*b^3)*d^7*e + 2040*(B*a^2*b + A*a*b^2)*d^6*e^2 - 2210*(B*a^3 + 3*A*a^2*b)
*d^5*e^3 + 429*(52*B*b^3*d*e^7 + 17*(3*B*a*b^2 + A*b^3)*e^8)*x^7 + 33*(802*B*b^3
*d^2*e^6 + 782*(3*B*a*b^2 + A*b^3)*d*e^7 + 765*(B*a^2*b + A*a*b^2)*e^8)*x^6 + 9*
(1212*B*b^3*d^3*e^5 + 3502*(3*B*a*b^2 + A*b^3)*d^2*e^6 + 10200*(B*a^2*b + A*a*b^
2)*d*e^7 + 1105*(B*a^3 + 3*A*a^2*b)*e^8)*x^5 + 5*(7*B*b^3*d^4*e^4 + 2431*A*a^3*e
^8 + 2720*(3*B*a*b^2 + A*b^3)*d^3*e^5 + 23358*(B*a^2*b + A*a*b^2)*d^2*e^6 + 7514
*(B*a^3 + 3*A*a^2*b)*d*e^7)*x^4 - 5*(8*B*b^3*d^5*e^3 - 9724*A*a^3*d*e^7 - 17*(3*
B*a*b^2 + A*b^3)*d^4*e^4 - 10812*(B*a^2*b + A*a*b^2)*d^3*e^5 - 10166*(B*a^3 + 3*
A*a^2*b)*d^2*e^6)*x^3 + 3*(16*B*b^3*d^6*e^2 + 24310*A*a^3*d^2*e^6 - 34*(3*B*a*b^
2 + A*b^3)*d^5*e^3 + 255*(B*a^2*b + A*a*b^2)*d^4*e^4 + 8840*(B*a^3 + 3*A*a^2*b)*
d^3*e^5)*x^2 - (64*B*b^3*d^7*e - 48620*A*a^3*d^3*e^5 - 136*(3*B*a*b^2 + A*b^3)*d
^6*e^2 + 1020*(B*a^2*b + A*a*b^2)*d^5*e^3 - 1105*(B*a^3 + 3*A*a^2*b)*d^4*e^4)*x)
*sqrt(e*x + d)/e^5

_______________________________________________________________________________________

Sympy [A]  time = 44.1309, size = 1523, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(7/2),x)

[Out]

Piecewise((2*A*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**3*d**3*x*sqrt(d + e*x)/9 +
4*A*a**3*d**2*e*x**2*sqrt(d + e*x)/3 + 8*A*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2
*A*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*A*a**2*b*d**5*sqrt(d + e*x)/(33*e**2) + 2*
A*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*A*a**2*b*d**3*x**2*sqrt(d + e*x)/11 +
92*A*a**2*b*d**2*e*x**3*sqrt(d + e*x)/33 + 68*A*a**2*b*d*e**2*x**4*sqrt(d + e*x)
/33 + 6*A*a**2*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*a*b**2*d**6*sqrt(d + e*x)/(42
9*e**3) - 8*A*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*A*a*b**2*d**4*x**2*sqrt
(d + e*x)/(143*e) + 424*A*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*A*a*b**2*d**2
*e*x**4*sqrt(d + e*x)/429 + 240*A*a*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 6*A*a*b
**2*e**3*x**6*sqrt(d + e*x)/13 - 32*A*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 16*A
*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*A*b**3*d**5*x**2*sqrt(d + e*x)/(2145*
e**2) + 2*A*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*A*b**3*d**3*x**4*sqrt(d
+ e*x)/1287 + 412*A*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 92*A*b**3*d*e**2*x**6*s
qrt(d + e*x)/195 + 2*A*b**3*e**3*x**7*sqrt(d + e*x)/15 - 4*B*a**3*d**5*sqrt(d +
e*x)/(99*e**2) + 2*B*a**3*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**3*d**3*x**2*sqrt
(d + e*x)/33 + 92*B*a**3*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**3*d*e**2*x**4*sq
rt(d + e*x)/99 + 2*B*a**3*e**3*x**5*sqrt(d + e*x)/11 + 16*B*a**2*b*d**6*sqrt(d +
e*x)/(429*e**3) - 8*B*a**2*b*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*B*a**2*b*d**4*
x**2*sqrt(d + e*x)/(143*e) + 424*B*a**2*b*d**3*x**3*sqrt(d + e*x)/429 + 916*B*a*
*2*b*d**2*e*x**4*sqrt(d + e*x)/429 + 240*B*a**2*b*d*e**2*x**5*sqrt(d + e*x)/143
+ 6*B*a**2*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*a*b**2*d**7*sqrt(d + e*x)/(2145*e
**4) + 16*B*a*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3) - 4*B*a*b**2*d**5*x**2*sqrt(
d + e*x)/(715*e**2) + 2*B*a*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 320*B*a*b**2*
d**3*x**4*sqrt(d + e*x)/429 + 1236*B*a*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B
*a*b**2*d*e**2*x**6*sqrt(d + e*x)/65 + 2*B*a*b**2*e**3*x**7*sqrt(d + e*x)/5 + 25
6*B*b**3*d**8*sqrt(d + e*x)/(109395*e**5) - 128*B*b**3*d**7*x*sqrt(d + e*x)/(109
395*e**4) + 32*B*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*B*b**3*d**5*x**3
*sqrt(d + e*x)/(21879*e**2) + 14*B*b**3*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424
*B*b**3*d**3*x**5*sqrt(d + e*x)/12155 + 1604*B*b**3*d**2*e*x**6*sqrt(d + e*x)/33
15 + 104*B*b**3*d*e**2*x**7*sqrt(d + e*x)/255 + 2*B*b**3*e**3*x**8*sqrt(d + e*x)
/17, Ne(e, 0)), (d**(7/2)*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3
*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b**2*x**4/4 + B*b**3*x**5/5), Tr
ue))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.258192, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Done