3.2165 \(\int (A+B x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=112 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (A b-a B) (a c+b c x)^{m+4}}{b^2 c^4 (m+4) (a+b x)}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a c+b c x)^{m+5}}{b^2 c^5 (m+5) (a+b x)} \]

[Out]

((A*b - a*B)*(a*c + b*c*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b^2*c^4*(4 +
m)*(a + b*x)) + (B*(a*c + b*c*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b^2*c^5
*(5 + m)*(a + b*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.214874, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (A b-a B) (a c+b c x)^{m+4}}{b^2 c^4 (m+4) (a+b x)}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2} (a c+b c x)^{m+5}}{b^2 c^5 (m+5) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(a*c + b*c*x)^(4 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b^2*c^4*(4 +
m)*(a + b*x)) + (B*(a*c + b*c*x)^(5 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(b^2*c^5
*(5 + m)*(a + b*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.6065, size = 90, normalized size = 0.8 \[ \frac{B \left (a c + b c x\right )^{m + 2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{b^{2} c^{2} \left (m + 5\right )} + \frac{\left (A b - B a\right ) \left (a c + b c x\right )^{m + 1} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{b^{2} c \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a*c + b*c*x)**(m + 2)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(b**2*c**2*(m + 5))
+ (A*b - B*a)*(a*c + b*c*x)**(m + 1)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(b**2*
c*(m + 4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.128543, size = 59, normalized size = 0.53 \[ \frac{(a+b x)^3 \sqrt{(a+b x)^2} (c (a+b x))^m (-a B+A b (m+5)+b B (m+4) x)}{b^2 (m+4) (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^3*(c*(a + b*x))^m*Sqrt[(a + b*x)^2]*(-(a*B) + A*b*(5 + m) + b*B*(4 +
m)*x))/(b^2*(4 + m)*(5 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 62, normalized size = 0.6 \[{\frac{ \left ( bxc+ac \right ) ^{m} \left ( Bbmx+Abm+4\,Bbx+5\,Ab-Ba \right ) \left ( bx+a \right ) }{{b}^{2} \left ({m}^{2}+9\,m+20 \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*x+a)^2)^(3/2)*(b*c*x+a*c)^m*(B*b*m*x+A*b*m+4*B*b*x+5*A*b-B*a)*(b*x+a)/b^2/(m
^2+9*m+20)

_______________________________________________________________________________________

Maxima [A]  time = 0.721986, size = 243, normalized size = 2.17 \[ \frac{{\left (b^{4} c^{m} x^{4} + 4 \, a b^{3} c^{m} x^{3} + 6 \, a^{2} b^{2} c^{m} x^{2} + 4 \, a^{3} b c^{m} x + a^{4} c^{m}\right )}{\left (b x + a\right )}^{m} A}{b{\left (m + 4\right )}} + \frac{{\left (b^{5} c^{m}{\left (m + 4\right )} x^{5} + a b^{4} c^{m}{\left (4 \, m + 15\right )} x^{4} + 2 \, a^{2} b^{3} c^{m}{\left (3 \, m + 10\right )} x^{3} + 2 \, a^{3} b^{2} c^{m}{\left (2 \, m + 5\right )} x^{2} + a^{4} b c^{m} m x - a^{5} c^{m}\right )}{\left (b x + a\right )}^{m} B}{{\left (m^{2} + 9 \, m + 20\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^4*c^m*x^4 + 4*a*b^3*c^m*x^3 + 6*a^2*b^2*c^m*x^2 + 4*a^3*b*c^m*x + a^4*c^m)*(b
*x + a)^m*A/(b*(m + 4)) + (b^5*c^m*(m + 4)*x^5 + a*b^4*c^m*(4*m + 15)*x^4 + 2*a^
2*b^3*c^m*(3*m + 10)*x^3 + 2*a^3*b^2*c^m*(2*m + 5)*x^2 + a^4*b*c^m*m*x - a^5*c^m
)*(b*x + a)^m*B/((m^2 + 9*m + 20)*b^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.327219, size = 296, normalized size = 2.64 \[ \frac{{\left (A a^{4} b m - B a^{5} + 5 \, A a^{4} b +{\left (B b^{5} m + 4 \, B b^{5}\right )} x^{5} +{\left (15 \, B a b^{4} + 5 \, A b^{5} +{\left (4 \, B a b^{4} + A b^{5}\right )} m\right )} x^{4} + 2 \,{\left (10 \, B a^{2} b^{3} + 10 \, A a b^{4} +{\left (3 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} m\right )} x^{3} + 2 \,{\left (5 \, B a^{3} b^{2} + 15 \, A a^{2} b^{3} +{\left (2 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} m\right )} x^{2} +{\left (20 \, A a^{3} b^{2} +{\left (B a^{4} b + 4 \, A a^{3} b^{2}\right )} m\right )} x\right )}{\left (b c x + a c\right )}^{m}}{b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*a^4*b*m - B*a^5 + 5*A*a^4*b + (B*b^5*m + 4*B*b^5)*x^5 + (15*B*a*b^4 + 5*A*b^5
+ (4*B*a*b^4 + A*b^5)*m)*x^4 + 2*(10*B*a^2*b^3 + 10*A*a*b^4 + (3*B*a^2*b^3 + 2*
A*a*b^4)*m)*x^3 + 2*(5*B*a^3*b^2 + 15*A*a^2*b^3 + (2*B*a^3*b^2 + 3*A*a^2*b^3)*m)
*x^2 + (20*A*a^3*b^2 + (B*a^4*b + 4*A*a^3*b^2)*m)*x)*(b*c*x + a*c)^m/(b^2*m^2 +
9*b^2*m + 20*b^2)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295866, size = 790, normalized size = 7.05 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b^5*m*x^5*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 4*B*a*b^4*m*x^4*e^(m*ln(b*c*x
+ a*c))*sign(b*x + a) + A*b^5*m*x^4*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 4*B*b
^5*x^5*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 6*B*a^2*b^3*m*x^3*e^(m*ln(b*c*x + a
*c))*sign(b*x + a) + 4*A*a*b^4*m*x^3*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 15*B*
a*b^4*x^4*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 5*A*b^5*x^4*e^(m*ln(b*c*x + a*c)
)*sign(b*x + a) + 4*B*a^3*b^2*m*x^2*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 6*A*a^
2*b^3*m*x^2*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 20*B*a^2*b^3*x^3*e^(m*ln(b*c*x
+ a*c))*sign(b*x + a) + 20*A*a*b^4*x^3*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + B*
a^4*b*m*x*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 4*A*a^3*b^2*m*x*e^(m*ln(b*c*x +
a*c))*sign(b*x + a) + 10*B*a^3*b^2*x^2*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + 30*
A*a^2*b^3*x^2*e^(m*ln(b*c*x + a*c))*sign(b*x + a) + A*a^4*b*m*e^(m*ln(b*c*x + a*
c))*sign(b*x + a) + 20*A*a^3*b^2*x*e^(m*ln(b*c*x + a*c))*sign(b*x + a) - B*a^5*e
^(m*ln(b*c*x + a*c))*sign(b*x + a) + 5*A*a^4*b*e^(m*ln(b*c*x + a*c))*sign(b*x +
a))/(b^2*m^2 + 9*b^2*m + 20*b^2)