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

Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]

[Out]

(2*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^2) + (2*B*(a + b*x)^(9/2))/(9*b^2)

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Rubi [A]  time = 0.044717, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac{2 B (a+b x)^{9/2}}{9 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(a + b*x)^(7/2))/(7*b^2) + (2*B*(a + b*x)^(9/2))/(9*b^2)

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Rubi in Sympy [A]  time = 7.92321, size = 37, normalized size = 0.88 \[ \frac{2 B \left (a + b x\right )^{\frac{9}{2}}}{9 b^{2}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x)**(9/2)/(9*b**2) + 2*(a + b*x)**(7/2)*(A*b - B*a)/(7*b**2)

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Mathematica [A]  time = 0.0489212, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} (-2 a B+9 A b+7 b B x)}{63 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(7/2)*(9*A*b - 2*a*B + 7*b*B*x))/(63*b^2)

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Maple [A]  time = 0.006, size = 27, normalized size = 0.6 \[{\frac{14\,bBx+18\,Ab-4\,Ba}{63\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(B*x+A),x)

[Out]

2/63*(b*x+a)^(7/2)*(7*B*b*x+9*A*b-2*B*a)/b^2

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Maxima [A]  time = 1.34411, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (7 \,{\left (b x + a\right )}^{\frac{9}{2}} B - 9 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{7}{2}}\right )}}{63 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*(b*x + a)^(9/2)*B - 9*(B*a - A*b)*(b*x + a)^(7/2))/b^2

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Fricas [A]  time = 0.207943, size = 126, normalized size = 3. \[ \frac{2 \,{\left (7 \, B b^{4} x^{4} - 2 \, B a^{4} + 9 \, A a^{3} b +{\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \,{\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{2} +{\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x\right )} \sqrt{b x + a}}{63 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(7*B*b^4*x^4 - 2*B*a^4 + 9*A*a^3*b + (19*B*a*b^3 + 9*A*b^4)*x^3 + 3*(5*B*a^
2*b^2 + 9*A*a*b^3)*x^2 + (B*a^3*b + 27*A*a^2*b^2)*x)*sqrt(b*x + a)/b^2

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Sympy [A]  time = 6.38216, size = 194, normalized size = 4.62 \[ \begin{cases} \frac{2 A a^{3} \sqrt{a + b x}}{7 b} + \frac{6 A a^{2} x \sqrt{a + b x}}{7} + \frac{6 A a b x^{2} \sqrt{a + b x}}{7} + \frac{2 A b^{2} x^{3} \sqrt{a + b x}}{7} - \frac{4 B a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 B a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 B a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 B a b x^{3} \sqrt{a + b x}}{63} + \frac{2 B b^{2} x^{4} \sqrt{a + b x}}{9} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (A x + \frac{B x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*A*a**3*sqrt(a + b*x)/(7*b) + 6*A*a**2*x*sqrt(a + b*x)/7 + 6*A*a*b*x
**2*sqrt(a + b*x)/7 + 2*A*b**2*x**3*sqrt(a + b*x)/7 - 4*B*a**4*sqrt(a + b*x)/(63
*b**2) + 2*B*a**3*x*sqrt(a + b*x)/(63*b) + 10*B*a**2*x**2*sqrt(a + b*x)/21 + 38*
B*a*b*x**3*sqrt(a + b*x)/63 + 2*B*b**2*x**4*sqrt(a + b*x)/9, Ne(b, 0)), (a**(5/2
)*(A*x + B*x**2/2), True))

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GIAC/XCAS [A]  time = 0.212739, size = 308, normalized size = 7.33 \[ \frac{2 \,{\left (105 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} + 42 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A a + \frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B a^{2}}{b} + \frac{6 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} B a}{b^{13}} + \frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} A}{b^{12}} + \frac{{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} B}{b^{25}}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(105*(b*x + a)^(3/2)*A*a^2 + 42*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*
A*a + 21*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*B*a^2/b + 6*(15*(b*x + a)^(7/
2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^12)*B*a/b^13 + 3*
(15*(b*x + a)^(7/2)*b^12 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^
12)*A/b^12 + (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)*B/b^25)/b