3.1890 \(\int (A+B x) (d+e x)^m \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=171 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2) (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+3}}{e^3 (m+3) (a+b x)} \]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(
1 + m)*(a + b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(e^3*(2 + m)*(a + b*x)) + (b*B*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(e^3*(3 + m)*(a + b*x))

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Rubi [A]  time = 0.276268, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1) (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2) (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{m+3}}{e^3 (m+3) (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^m*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^3*(
1 + m)*(a + b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(2 + m)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(e^3*(2 + m)*(a + b*x)) + (b*B*(d + e*x)^(3 + m)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])/(e^3*(3 + m)*(a + b*x))

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Rubi in Sympy [A]  time = 38.8398, size = 187, normalized size = 1.09 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{m + 1} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 b e \left (m + 3\right )} - \frac{\left (d + e x\right )^{m + 1} \left (- A b e \left (m + 3\right ) + B \left (a e \left (m + 1\right ) + 2 b d\right )\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b e^{2} \left (m + 2\right ) \left (m + 3\right )} - \frac{\left (d + e x\right )^{m + 1} \left (a e - b d\right ) \left (- A b e \left (m + 3\right ) + B \left (a e \left (m + 1\right ) + 2 b d\right )\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b e^{3} \left (a + b x\right ) \left (m + 1\right ) \left (m + 2\right ) \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(1/2),x)

[Out]

B*(2*a + 2*b*x)*(d + e*x)**(m + 1)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2*b*e*(m +
3)) - (d + e*x)**(m + 1)*(-A*b*e*(m + 3) + B*(a*e*(m + 1) + 2*b*d))*sqrt(a**2 +
2*a*b*x + b**2*x**2)/(b*e**2*(m + 2)*(m + 3)) - (d + e*x)**(m + 1)*(a*e - b*d)*(
-A*b*e*(m + 3) + B*(a*e*(m + 1) + 2*b*d))*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(b*e*
*3*(a + b*x)*(m + 1)*(m + 2)*(m + 3))

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Mathematica [A]  time = 0.193417, size = 121, normalized size = 0.71 \[ \frac{\sqrt{(a+b x)^2} (d+e x)^{m+1} \left (a e (m+3) (A e (m+2)-B d+B e (m+1) x)+b \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )\right )}{e^3 (m+1) (m+2) (m+3) (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^m*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(Sqrt[(a + b*x)^2]*(d + e*x)^(1 + m)*(a*e*(3 + m)*(-(B*d) + A*e*(2 + m) + B*e*(1
+ m)*x) + b*(A*e*(3 + m)*(-d + e*(1 + m)*x) + B*(2*d^2 - 2*d*e*(1 + m)*x + e^2*
(2 + 3*m + m^2)*x^2))))/(e^3*(1 + m)*(2 + m)*(3 + m)*(a + b*x))

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Maple [A]  time = 0.01, size = 205, normalized size = 1.2 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bb{e}^{2}{m}^{2}{x}^{2}+Ab{e}^{2}{m}^{2}x+Ba{e}^{2}{m}^{2}x+3\,Bb{e}^{2}m{x}^{2}+Aa{e}^{2}{m}^{2}+4\,Ab{e}^{2}mx+4\,Ba{e}^{2}mx-2\,Bbdemx+2\,B{x}^{2}b{e}^{2}+5\,Aa{e}^{2}m-Abdem+3\,Ab{e}^{2}x-aBdem+3\,aB{e}^{2}x-2\,Bbdex+6\,A{e}^{2}a-3\,Abde-3\,aBde+2\,Bb{d}^{2} \right ) }{ \left ( bx+a \right ){e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^m*(b^2*x^2+2*a*b*x+a^2)^(1/2),x)

[Out]

(e*x+d)^(1+m)*(B*b*e^2*m^2*x^2+A*b*e^2*m^2*x+B*a*e^2*m^2*x+3*B*b*e^2*m*x^2+A*a*e
^2*m^2+4*A*b*e^2*m*x+4*B*a*e^2*m*x-2*B*b*d*e*m*x+2*B*b*e^2*x^2+5*A*a*e^2*m-A*b*d
*e*m+3*A*b*e^2*x-B*a*d*e*m+3*B*a*e^2*x-2*B*b*d*e*x+6*A*a*e^2-3*A*b*d*e-3*B*a*d*e
+2*B*b*d^2)*((b*x+a)^2)^(1/2)/(b*x+a)/e^3/(m^3+6*m^2+11*m+6)

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Maxima [A]  time = 0.731297, size = 239, normalized size = 1.4 \[ \frac{{\left (b e^{2}{\left (m + 1\right )} x^{2} + a d e{\left (m + 2\right )} - b d^{2} +{\left (a e^{2}{\left (m + 2\right )} + b d e m\right )} x\right )}{\left (e x + d\right )}^{m} A}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac{{\left ({\left (m^{2} + 3 \, m + 2\right )} b e^{3} x^{3} - a d^{2} e{\left (m + 3\right )} + 2 \, b d^{3} +{\left ({\left (m^{2} + m\right )} b d e^{2} +{\left (m^{2} + 4 \, m + 3\right )} a e^{3}\right )} x^{2} +{\left ({\left (m^{2} + 3 \, m\right )} a d e^{2} - 2 \, b d^{2} e m\right )} x\right )}{\left (e x + d\right )}^{m} B}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^m,x, algorithm="maxima")

[Out]

(b*e^2*(m + 1)*x^2 + a*d*e*(m + 2) - b*d^2 + (a*e^2*(m + 2) + b*d*e*m)*x)*(e*x +
d)^m*A/((m^2 + 3*m + 2)*e^2) + ((m^2 + 3*m + 2)*b*e^3*x^3 - a*d^2*e*(m + 3) + 2
*b*d^3 + ((m^2 + m)*b*d*e^2 + (m^2 + 4*m + 3)*a*e^3)*x^2 + ((m^2 + 3*m)*a*d*e^2
- 2*b*d^2*e*m)*x)*(e*x + d)^m*B/((m^3 + 6*m^2 + 11*m + 6)*e^3)

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Fricas [A]  time = 0.32029, size = 346, normalized size = 2.02 \[ \frac{{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \,{\left (B a + A b\right )} d^{2} e +{\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} +{\left (3 \,{\left (B a + A b\right )} e^{3} +{\left (B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} m^{2} +{\left (B b d e^{2} + 4 \,{\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} +{\left (5 \, A a d e^{2} -{\left (B a + A b\right )} d^{2} e\right )} m +{\left (6 \, A a e^{3} +{\left (A a e^{3} +{\left (B a + A b\right )} d e^{2}\right )} m^{2} -{\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \,{\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(A*a*d*e^2*m^2 + 2*B*b*d^3 + 6*A*a*d*e^2 - 3*(B*a + A*b)*d^2*e + (B*b*e^3*m^2 +
3*B*b*e^3*m + 2*B*b*e^3)*x^3 + (3*(B*a + A*b)*e^3 + (B*b*d*e^2 + (B*a + A*b)*e^3
)*m^2 + (B*b*d*e^2 + 4*(B*a + A*b)*e^3)*m)*x^2 + (5*A*a*d*e^2 - (B*a + A*b)*d^2*
e)*m + (6*A*a*e^3 + (A*a*e^3 + (B*a + A*b)*d*e^2)*m^2 - (2*B*b*d^2*e - 5*A*a*e^3
- 3*(B*a + A*b)*d*e^2)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^
3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (A + B x\right ) \left (d + e x\right )^{m} \sqrt{\left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**m*(b**2*x**2+2*a*b*x+a**2)**(1/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**m*sqrt((a + b*x)**2), x)

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GIAC/XCAS [A]  time = 0.302498, size = 963, normalized size = 5.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b^2*x^2 + 2*a*b*x + a^2)*(B*x + A)*(e*x + d)^m,x, algorithm="giac")

[Out]

(B*b*m^2*x^3*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + B*b*d*m^2*x^2*e^(m*ln(x*e + d
) + 2)*sign(b*x + a) + B*a*m^2*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + A*b*m^2
*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 3*B*b*m*x^3*e^(m*ln(x*e + d) + 3)*sig
n(b*x + a) + B*a*d*m^2*x*e^(m*ln(x*e + d) + 2)*sign(b*x + a) + A*b*d*m^2*x*e^(m*
ln(x*e + d) + 2)*sign(b*x + a) + B*b*d*m*x^2*e^(m*ln(x*e + d) + 2)*sign(b*x + a)
- 2*B*b*d^2*m*x*e^(m*ln(x*e + d) + 1)*sign(b*x + a) + A*a*m^2*x*e^(m*ln(x*e + d
) + 3)*sign(b*x + a) + 4*B*a*m*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 4*A*b*m
*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 2*B*b*x^3*e^(m*ln(x*e + d) + 3)*sign(
b*x + a) + A*a*d*m^2*e^(m*ln(x*e + d) + 2)*sign(b*x + a) + 3*B*a*d*m*x*e^(m*ln(x
*e + d) + 2)*sign(b*x + a) + 3*A*b*d*m*x*e^(m*ln(x*e + d) + 2)*sign(b*x + a) - B
*a*d^2*m*e^(m*ln(x*e + d) + 1)*sign(b*x + a) - A*b*d^2*m*e^(m*ln(x*e + d) + 1)*s
ign(b*x + a) + 2*B*b*d^3*e^(m*ln(x*e + d))*sign(b*x + a) + 5*A*a*m*x*e^(m*ln(x*e
+ d) + 3)*sign(b*x + a) + 3*B*a*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 3*A*b
*x^2*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 5*A*a*d*m*e^(m*ln(x*e + d) + 2)*sign(
b*x + a) - 3*B*a*d^2*e^(m*ln(x*e + d) + 1)*sign(b*x + a) - 3*A*b*d^2*e^(m*ln(x*e
+ d) + 1)*sign(b*x + a) + 6*A*a*x*e^(m*ln(x*e + d) + 3)*sign(b*x + a) + 6*A*a*d
*e^(m*ln(x*e + d) + 2)*sign(b*x + a))/(m^3*e^3 + 6*m^2*e^3 + 11*m*e^3 + 6*e^3)