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

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]

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

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {2 (a+b x)^{7/2} (A b-a B)}{7 b^2}+\frac {2 B (a+b x)^{9/2}}{9 b^2} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x)^{5/2} (A+B x) \, dx &=\int \left (\frac {(A b-a B) (a+b x)^{5/2}}{b}+\frac {B (a+b x)^{7/2}}{b}\right ) \, dx\\ &=\frac {2 (A b-a B) (a+b x)^{7/2}}{7 b^2}+\frac {2 B (a+b x)^{9/2}}{9 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.03, 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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fricas [B]  time = 0.66, size = 93, normalized size = 2.21 \[ \frac {2 \, {\left (7 \, B b^{4} x^{4} - 2 \, B a^{4} + 9 \, A a^{3} b + {\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{3} + 3 \, {\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{2} + {\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x + a}}{63 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A),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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giac [B]  time = 1.28, size = 306, normalized size = 7.29 \[ \frac {2 \, {\left (315 \, \sqrt {b x + a} A a^{3} + 315 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} A a^{2} + \frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} B a^{3}}{b} + 63 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} A a + \frac {63 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} B a^{2}}{b} + 9 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A + \frac {27 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} B a}{b} + \frac {{\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B}{b}\right )}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(315*sqrt(b*x + a)*A*a^3 + 315*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*A*a^2 + 105*((b*x + a)^(3/2) - 3*sq
rt(b*x + a)*a)*B*a^3/b + 63*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*A*a + 63*(3*(b*x
+ a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*B*a^2/b + 9*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)
*a + 35*(b*x + a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*A + 27*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x
+ a)^(3/2)*a^2 - 35*sqrt(b*x + a)*a^3)*B*a/b + (35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5
/2)*a^2 - 420*(b*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*B/b)/b

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maple [A]  time = 0.00, size = 27, normalized size = 0.64 \[ \frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (7 B b x +9 A b -2 B a \right )}{63 b^{2}} \]

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 = 0.89, size = 33, normalized size = 0.79 \[ \frac {2 \, {\left (7 \, {\left (b x + a\right )}^{\frac {9}{2}} B - 9 \, {\left (B a - A b\right )} {\left (b x + a\right )}^{\frac {7}{2}}\right )}}{63 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A),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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mupad [B]  time = 0.04, size = 29, normalized size = 0.69 \[ \frac {2\,{\left (a+b\,x\right )}^{7/2}\,\left (9\,A\,b-9\,B\,a+7\,B\,\left (a+b\,x\right )\right )}{63\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^(5/2),x)

[Out]

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

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sympy [A]  time = 2.25, 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*sq
rt(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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