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

Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (a+b x)^{7/2}}{7 b^2} \]

[Out]

(2*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^2) + (2*B*(a + b*x)^(7/2))/(7*b^2)

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Rubi [A]  time = 0.0446655, 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 \[ \frac{2 (a+b x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (a+b x)^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^2) + (2*B*(a + b*x)^(7/2))/(7*b^2)

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Rubi in Sympy [A]  time = 7.37228, size = 37, normalized size = 0.88 \[ \frac{2 B \left (a + b x\right )^{\frac{7}{2}}}{7 b^{2}} + \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(a + b*x)**(7/2)/(7*b**2) + 2*(a + b*x)**(5/2)*(A*b - B*a)/(5*b**2)

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Mathematica [A]  time = 0.0381289, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{5/2} (-2 a B+7 A b+5 b B x)}{35 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(5/2)*(7*A*b - 2*a*B + 5*b*B*x))/(35*b^2)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \[{\frac{10\,bBx+14\,Ab-4\,Ba}{35\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A),x)

[Out]

2/35*(b*x+a)^(5/2)*(5*B*b*x+7*A*b-2*B*a)/b^2

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Maxima [A]  time = 1.41908, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 7 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{35 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*(b*x + a)^(7/2)*B - 7*(B*a - A*b)*(b*x + a)^(5/2))/b^2

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Fricas [A]  time = 0.204942, size = 93, normalized size = 2.21 \[ \frac{2 \,{\left (5 \, B b^{3} x^{3} - 2 \, B a^{3} + 7 \, A a^{2} b +{\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} +{\left (B a^{2} b + 14 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{35 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*B*b^3*x^3 - 2*B*a^3 + 7*A*a^2*b + (8*B*a*b^2 + 7*A*b^3)*x^2 + (B*a^2*b +
14*A*a*b^2)*x)*sqrt(b*x + a)/b^2

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Sympy [A]  time = 1.88818, size = 146, normalized size = 3.48 \[ \begin{cases} \frac{2 A a^{2} \sqrt{a + b x}}{5 b} + \frac{4 A a x \sqrt{a + b x}}{5} + \frac{2 A b x^{2} \sqrt{a + b x}}{5} - \frac{4 B a^{3} \sqrt{a + b x}}{35 b^{2}} + \frac{2 B a^{2} x \sqrt{a + b x}}{35 b} + \frac{16 B a x^{2} \sqrt{a + b x}}{35} + \frac{2 B b x^{3} \sqrt{a + b x}}{7} & \text{for}\: b \neq 0 \\a^{\frac{3}{2}} \left (A x + \frac{B x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*A*a**2*sqrt(a + b*x)/(5*b) + 4*A*a*x*sqrt(a + b*x)/5 + 2*A*b*x**2*s
qrt(a + b*x)/5 - 4*B*a**3*sqrt(a + b*x)/(35*b**2) + 2*B*a**2*x*sqrt(a + b*x)/(35
*b) + 16*B*a*x**2*sqrt(a + b*x)/35 + 2*B*b*x**3*sqrt(a + b*x)/7, Ne(b, 0)), (a**
(3/2)*(A*x + B*x**2/2), True))

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GIAC/XCAS [A]  time = 0.208919, size = 153, normalized size = 3.64 \[ \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{3}{2}} A a + 7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A + \frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B a}{b} + \frac{{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}\right )} B}{b^{13}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(35*(b*x + a)^(3/2)*A*a + 7*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*A +
7*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*B*a/b + (15*(b*x + a)^(7/2)*b^12 - 4
2*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^12)*B/b^13)/b