3.399 \(\int x^3 (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=122 \[ -\frac{2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac{2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac{2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac{6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac{2 B (a+b x)^{15/2}}{15 b^5} \]

[Out]

(-2*a^3*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(9/2))/(9*b^5) - (6*a*(A*b - 2*a*B)*(a + b*x)^(11/2))/(11*b^5) + (2*(A*b - 4*a*B
)*(a + b*x)^(13/2))/(13*b^5) + (2*B*(a + b*x)^(15/2))/(15*b^5)

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Rubi [A]  time = 0.156884, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a^3 (a+b x)^{7/2} (A b-a B)}{7 b^5}+\frac{2 a^2 (a+b x)^{9/2} (3 A b-4 a B)}{9 b^5}+\frac{2 (a+b x)^{13/2} (A b-4 a B)}{13 b^5}-\frac{6 a (a+b x)^{11/2} (A b-2 a B)}{11 b^5}+\frac{2 B (a+b x)^{15/2}}{15 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(-2*a^3*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(9/2))/(9*b^5) - (6*a*(A*b - 2*a*B)*(a + b*x)^(11/2))/(11*b^5) + (2*(A*b - 4*a*B
)*(a + b*x)^(13/2))/(13*b^5) + (2*B*(a + b*x)^(15/2))/(15*b^5)

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Rubi in Sympy [A]  time = 27.1526, size = 119, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{15}{2}}}{15 b^{5}} - \frac{2 a^{3} \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{5}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{9}{2}} \left (3 A b - 4 B a\right )}{9 b^{5}} - \frac{6 a \left (a + b x\right )^{\frac{11}{2}} \left (A b - 2 B a\right )}{11 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{13}{2}} \left (A b - 4 B a\right )}{13 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*B*(a + b*x)**(15/2)/(15*b**5) - 2*a**3*(a + b*x)**(7/2)*(A*b - B*a)/(7*b**5) +
2*a**2*(a + b*x)**(9/2)*(3*A*b - 4*B*a)/(9*b**5) - 6*a*(a + b*x)**(11/2)*(A*b -
2*B*a)/(11*b**5) + 2*(a + b*x)**(13/2)*(A*b - 4*B*a)/(13*b**5)

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Mathematica [A]  time = 0.0945166, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} \left (128 a^4 B-16 a^3 b (15 A+28 B x)+168 a^2 b^2 x (5 A+6 B x)-42 a b^3 x^2 (45 A+44 B x)+231 b^4 x^3 (15 A+13 B x)\right )}{45045 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(7/2)*(128*a^4*B + 168*a^2*b^2*x*(5*A + 6*B*x) + 231*b^4*x^3*(15*A
+ 13*B*x) - 16*a^3*b*(15*A + 28*B*x) - 42*a*b^3*x^2*(45*A + 44*B*x)))/(45045*b^5
)

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Maple [A]  time = 0.009, size = 95, normalized size = 0.8 \[ -{\frac{-6006\,B{x}^{4}{b}^{4}-6930\,A{b}^{4}{x}^{3}+3696\,Ba{b}^{3}{x}^{3}+3780\,Aa{b}^{3}{x}^{2}-2016\,B{a}^{2}{b}^{2}{x}^{2}-1680\,A{a}^{2}{b}^{2}x+896\,B{a}^{3}bx+480\,A{a}^{3}b-256\,B{a}^{4}}{45045\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

-2/45045*(b*x+a)^(7/2)*(-3003*B*b^4*x^4-3465*A*b^4*x^3+1848*B*a*b^3*x^3+1890*A*a
*b^3*x^2-1008*B*a^2*b^2*x^2-840*A*a^2*b^2*x+448*B*a^3*b*x+240*A*a^3*b-128*B*a^4)
/b^5

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Maxima [A]  time = 1.36065, size = 135, normalized size = 1.11 \[ \frac{2 \,{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} B - 3465 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{13}{2}} + 12285 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{11}{2}} - 5005 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 6435 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{45045 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="maxima")

[Out]

2/45045*(3003*(b*x + a)^(15/2)*B - 3465*(4*B*a - A*b)*(b*x + a)^(13/2) + 12285*(
2*B*a^2 - A*a*b)*(b*x + a)^(11/2) - 5005*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(9/2) +
6435*(B*a^4 - A*a^3*b)*(b*x + a)^(7/2))/b^5

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Fricas [A]  time = 0.207357, size = 225, normalized size = 1.84 \[ \frac{2 \,{\left (3003 \, B b^{7} x^{7} + 128 \, B a^{7} - 240 \, A a^{6} b + 231 \,{\left (31 \, B a b^{6} + 15 \, A b^{7}\right )} x^{6} + 63 \,{\left (71 \, B a^{2} b^{5} + 135 \, A a b^{6}\right )} x^{5} + 35 \,{\left (B a^{3} b^{4} + 159 \, A a^{2} b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{4} b^{3} - 15 \, A a^{3} b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{6} b - 15 \, A a^{5} b^{2}\right )} x\right )} \sqrt{b x + a}}{45045 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="fricas")

[Out]

2/45045*(3003*B*b^7*x^7 + 128*B*a^7 - 240*A*a^6*b + 231*(31*B*a*b^6 + 15*A*b^7)*
x^6 + 63*(71*B*a^2*b^5 + 135*A*a*b^6)*x^5 + 35*(B*a^3*b^4 + 159*A*a^2*b^5)*x^4 -
5*(8*B*a^4*b^3 - 15*A*a^3*b^4)*x^3 + 6*(8*B*a^5*b^2 - 15*A*a^4*b^3)*x^2 - 8*(8*
B*a^6*b - 15*A*a^5*b^2)*x)*sqrt(b*x + a)/b^5

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Sympy [A]  time = 8.728, size = 496, normalized size = 4.07 \[ \frac{2 A a^{2} \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}} + \frac{4 A a \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{4}} + \frac{2 A \left (- \frac{a^{5} \left (a + b x\right )^{\frac{3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac{5}{2}} - \frac{10 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{10 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{9} - \frac{5 a \left (a + b x\right )^{\frac{11}{2}}}{11} + \frac{\left (a + b x\right )^{\frac{13}{2}}}{13}\right )}{b^{4}} + \frac{2 B a^{2} \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{5}} + \frac{4 B a \left (- \frac{a^{5} \left (a + b x\right )^{\frac{3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac{5}{2}} - \frac{10 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{10 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{9} - \frac{5 a \left (a + b x\right )^{\frac{11}{2}}}{11} + \frac{\left (a + b x\right )^{\frac{13}{2}}}{13}\right )}{b^{5}} + \frac{2 B \left (\frac{a^{6} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{6 a^{5} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{15 a^{4} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{20 a^{3} \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{15 a^{2} \left (a + b x\right )^{\frac{11}{2}}}{11} - \frac{6 a \left (a + b x\right )^{\frac{13}{2}}}{13} + \frac{\left (a + b x\right )^{\frac{15}{2}}}{15}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*A*a**2*(-a**3*(a + b*x)**(3/2)/3 + 3*a**2*(a + b*x)**(5/2)/5 - 3*a*(a + b*x)**
(7/2)/7 + (a + b*x)**(9/2)/9)/b**4 + 4*A*a*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a
+ b*x)**(5/2)/5 + 6*a**2*(a + b*x)**(7/2)/7 - 4*a*(a + b*x)**(9/2)/9 + (a + b*x)
**(11/2)/11)/b**4 + 2*A*(-a**5*(a + b*x)**(3/2)/3 + a**4*(a + b*x)**(5/2) - 10*a
**3*(a + b*x)**(7/2)/7 + 10*a**2*(a + b*x)**(9/2)/9 - 5*a*(a + b*x)**(11/2)/11 +
(a + b*x)**(13/2)/13)/b**4 + 2*B*a**2*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a + b*
x)**(5/2)/5 + 6*a**2*(a + b*x)**(7/2)/7 - 4*a*(a + b*x)**(9/2)/9 + (a + b*x)**(1
1/2)/11)/b**5 + 4*B*a*(-a**5*(a + b*x)**(3/2)/3 + a**4*(a + b*x)**(5/2) - 10*a**
3*(a + b*x)**(7/2)/7 + 10*a**2*(a + b*x)**(9/2)/9 - 5*a*(a + b*x)**(11/2)/11 + (
a + b*x)**(13/2)/13)/b**5 + 2*B*(a**6*(a + b*x)**(3/2)/3 - 6*a**5*(a + b*x)**(5/
2)/5 + 15*a**4*(a + b*x)**(7/2)/7 - 20*a**3*(a + b*x)**(9/2)/9 + 15*a**2*(a + b*
x)**(11/2)/11 - 6*a*(a + b*x)**(13/2)/13 + (a + b*x)**(15/2)/15)/b**5

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GIAC/XCAS [A]  time = 0.24979, size = 702, normalized size = 5.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^3,x, algorithm="giac")

[Out]

2/45045*(143*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 189*(b*x +
a)^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*A*a^2/b^27 + 13*(315*(b*x + a)
^(11/2)*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 277
2*(b*x + a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*B*a^2/b^44 + 26*(315
*(b*x + a)^(11/2)*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*
b^40 - 2772*(b*x + a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*A*a/b^43 +
10*(693*(b*x + a)^(13/2)*b^60 - 4095*(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^
(9/2)*a^2*b^60 - 12870*(b*x + a)^(7/2)*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60
- 3003*(b*x + a)^(3/2)*a^5*b^60)*B*a/b^64 + 5*(693*(b*x + a)^(13/2)*b^60 - 4095*
(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^(9/2)*a^2*b^60 - 12870*(b*x + a)^(7/2)
*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60 - 3003*(b*x + a)^(3/2)*a^5*b^60)*A/b^6
3 + (3003*(b*x + a)^(15/2)*b^84 - 20790*(b*x + a)^(13/2)*a*b^84 + 61425*(b*x + a
)^(11/2)*a^2*b^84 - 100100*(b*x + a)^(9/2)*a^3*b^84 + 96525*(b*x + a)^(7/2)*a^4*
b^84 - 54054*(b*x + a)^(5/2)*a^5*b^84 + 15015*(b*x + a)^(3/2)*a^6*b^84)*B/b^88)/
b