3.398 \(\int x^4 (a+b x)^{5/2} (A+B x) \, dx\)

Optimal. Leaf size=151 \[ \frac{2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}-\frac{2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac{4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}+\frac{2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac{4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac{2 B (a+b x)^{17/2}}{17 b^6} \]

[Out]

(2*a^4*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^6) - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(
9/2))/(9*b^6) + (4*a^2*(3*A*b - 5*a*B)*(a + b*x)^(11/2))/(11*b^6) - (4*a*(2*A*b
- 5*a*B)*(a + b*x)^(13/2))/(13*b^6) + (2*(A*b - 5*a*B)*(a + b*x)^(15/2))/(15*b^6
) + (2*B*(a + b*x)^(17/2))/(17*b^6)

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Rubi [A]  time = 0.184901, antiderivative size = 151, 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 \[ \frac{2 a^4 (a+b x)^{7/2} (A b-a B)}{7 b^6}-\frac{2 a^3 (a+b x)^{9/2} (4 A b-5 a B)}{9 b^6}+\frac{4 a^2 (a+b x)^{11/2} (3 A b-5 a B)}{11 b^6}+\frac{2 (a+b x)^{15/2} (A b-5 a B)}{15 b^6}-\frac{4 a (a+b x)^{13/2} (2 A b-5 a B)}{13 b^6}+\frac{2 B (a+b x)^{17/2}}{17 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*a^4*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^6) - (2*a^3*(4*A*b - 5*a*B)*(a + b*x)^(
9/2))/(9*b^6) + (4*a^2*(3*A*b - 5*a*B)*(a + b*x)^(11/2))/(11*b^6) - (4*a*(2*A*b
- 5*a*B)*(a + b*x)^(13/2))/(13*b^6) + (2*(A*b - 5*a*B)*(a + b*x)^(15/2))/(15*b^6
) + (2*B*(a + b*x)^(17/2))/(17*b^6)

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Rubi in Sympy [A]  time = 28.8968, size = 150, normalized size = 0.99 \[ \frac{2 B \left (a + b x\right )^{\frac{17}{2}}}{17 b^{6}} + \frac{2 a^{4} \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{6}} - \frac{2 a^{3} \left (a + b x\right )^{\frac{9}{2}} \left (4 A b - 5 B a\right )}{9 b^{6}} + \frac{4 a^{2} \left (a + b x\right )^{\frac{11}{2}} \left (3 A b - 5 B a\right )}{11 b^{6}} - \frac{4 a \left (a + b x\right )^{\frac{13}{2}} \left (2 A b - 5 B a\right )}{13 b^{6}} + \frac{2 \left (a + b x\right )^{\frac{15}{2}} \left (A b - 5 B a\right )}{15 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*B*(a + b*x)**(17/2)/(17*b**6) + 2*a**4*(a + b*x)**(7/2)*(A*b - B*a)/(7*b**6) -
2*a**3*(a + b*x)**(9/2)*(4*A*b - 5*B*a)/(9*b**6) + 4*a**2*(a + b*x)**(11/2)*(3*
A*b - 5*B*a)/(11*b**6) - 4*a*(a + b*x)**(13/2)*(2*A*b - 5*B*a)/(13*b**6) + 2*(a
+ b*x)**(15/2)*(A*b - 5*B*a)/(15*b**6)

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Mathematica [A]  time = 0.133344, size = 106, normalized size = 0.7 \[ \frac{2 (a+b x)^{7/2} \left (-1280 a^5 B+128 a^4 b (17 A+35 B x)-224 a^3 b^2 x (34 A+45 B x)+336 a^2 b^3 x^2 (51 A+55 B x)-462 a b^4 x^3 (68 A+65 B x)+3003 b^5 x^4 (17 A+15 B x)\right )}{765765 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x)^(5/2)*(A + B*x),x]

[Out]

(2*(a + b*x)^(7/2)*(-1280*a^5*B + 3003*b^5*x^4*(17*A + 15*B*x) + 128*a^4*b*(17*A
+ 35*B*x) - 224*a^3*b^2*x*(34*A + 45*B*x) + 336*a^2*b^3*x^2*(51*A + 55*B*x) - 4
62*a*b^4*x^3*(68*A + 65*B*x)))/(765765*b^6)

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Maple [A]  time = 0.008, size = 119, normalized size = 0.8 \[{\frac{90090\,{b}^{5}B{x}^{5}+102102\,A{x}^{4}{b}^{5}-60060\,B{x}^{4}a{b}^{4}-62832\,A{x}^{3}a{b}^{4}+36960\,B{x}^{3}{a}^{2}{b}^{3}+34272\,A{x}^{2}{a}^{2}{b}^{3}-20160\,B{x}^{2}{a}^{3}{b}^{2}-15232\,Ax{a}^{3}{b}^{2}+8960\,Bx{a}^{4}b+4352\,A{a}^{4}b-2560\,B{a}^{5}}{765765\,{b}^{6}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)^(5/2)*(B*x+A),x)

[Out]

2/765765*(b*x+a)^(7/2)*(45045*B*b^5*x^5+51051*A*b^5*x^4-30030*B*a*b^4*x^4-31416*
A*a*b^4*x^3+18480*B*a^2*b^3*x^3+17136*A*a^2*b^3*x^2-10080*B*a^3*b^2*x^2-7616*A*a
^3*b^2*x+4480*B*a^4*b*x+2176*A*a^4*b-1280*B*a^5)/b^6

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Maxima [A]  time = 1.35449, size = 166, normalized size = 1.1 \[ \frac{2 \,{\left (45045 \,{\left (b x + a\right )}^{\frac{17}{2}} B - 51051 \,{\left (5 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{15}{2}} + 117810 \,{\left (5 \, B a^{2} - 2 \, A a b\right )}{\left (b x + a\right )}^{\frac{13}{2}} - 139230 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{11}{2}} + 85085 \,{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\left (b x + a\right )}^{\frac{9}{2}} - 109395 \,{\left (B a^{5} - A a^{4} b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{765765 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^4,x, algorithm="maxima")

[Out]

2/765765*(45045*(b*x + a)^(17/2)*B - 51051*(5*B*a - A*b)*(b*x + a)^(15/2) + 1178
10*(5*B*a^2 - 2*A*a*b)*(b*x + a)^(13/2) - 139230*(5*B*a^3 - 3*A*a^2*b)*(b*x + a)
^(11/2) + 85085*(5*B*a^4 - 4*A*a^3*b)*(b*x + a)^(9/2) - 109395*(B*a^5 - A*a^4*b)
*(b*x + a)^(7/2))/b^6

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Fricas [A]  time = 0.214363, size = 259, normalized size = 1.72 \[ \frac{2 \,{\left (45045 \, B b^{8} x^{8} - 1280 \, B a^{8} + 2176 \, A a^{7} b + 3003 \,{\left (35 \, B a b^{7} + 17 \, A b^{8}\right )} x^{7} + 231 \,{\left (275 \, B a^{2} b^{6} + 527 \, A a b^{7}\right )} x^{6} + 63 \,{\left (5 \, B a^{3} b^{5} + 1207 \, A a^{2} b^{6}\right )} x^{5} - 35 \,{\left (10 \, B a^{4} b^{4} - 17 \, A a^{3} b^{5}\right )} x^{4} + 40 \,{\left (10 \, B a^{5} b^{3} - 17 \, A a^{4} b^{4}\right )} x^{3} - 48 \,{\left (10 \, B a^{6} b^{2} - 17 \, A a^{5} b^{3}\right )} x^{2} + 64 \,{\left (10 \, B a^{7} b - 17 \, A a^{6} b^{2}\right )} x\right )} \sqrt{b x + a}}{765765 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(5/2)*x^4,x, algorithm="fricas")

[Out]

2/765765*(45045*B*b^8*x^8 - 1280*B*a^8 + 2176*A*a^7*b + 3003*(35*B*a*b^7 + 17*A*
b^8)*x^7 + 231*(275*B*a^2*b^6 + 527*A*a*b^7)*x^6 + 63*(5*B*a^3*b^5 + 1207*A*a^2*
b^6)*x^5 - 35*(10*B*a^4*b^4 - 17*A*a^3*b^5)*x^4 + 40*(10*B*a^5*b^3 - 17*A*a^4*b^
4)*x^3 - 48*(10*B*a^6*b^2 - 17*A*a^5*b^3)*x^2 + 64*(10*B*a^7*b - 17*A*a^6*b^2)*x
)*sqrt(b*x + a)/b^6

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Sympy [A]  time = 9.33036, size = 586, normalized size = 3.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)**(5/2)*(B*x+A),x)

[Out]

2*A*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)**(11/2)/11)/b**5 + 4*A*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*
A*(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 + 2*B*a**2*(-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**6 + 4*B*a*(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**6 + 2*B*(-a**7*(a + b*x)**(3/2)/3 + 7*a**6*(a + b*x)**(5/2)/5
- 3*a**5*(a + b*x)**(7/2) + 35*a**4*(a + b*x)**(9/2)/9 - 35*a**3*(a + b*x)**(11
/2)/11 + 21*a**2*(a + b*x)**(13/2)/13 - 7*a*(a + b*x)**(15/2)/15 + (a + b*x)**(1
7/2)/17)/b**6

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GIAC/XCAS [A]  time = 0.217592, 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)^(5/2)*x^4,x, algorithm="giac")

[Out]

Done