3.977 \(\int (d x)^m \sqrt{c x^2} (a+b x)^2 \, dx\)

Optimal. Leaf size=94 \[ \frac{a^2 \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{2 a b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac{b^2 \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

[Out]

(a^2*(d*x)^(2 + m)*Sqrt[c*x^2])/(d^2*(2 + m)*x) + (2*a*b*(d*x)^(3 + m)*Sqrt[c*x^
2])/(d^3*(3 + m)*x) + (b^2*(d*x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*x)

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Rubi [A]  time = 0.0980569, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a^2 \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{2 a b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac{b^2 \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(d*x)^(2 + m)*Sqrt[c*x^2])/(d^2*(2 + m)*x) + (2*a*b*(d*x)^(3 + m)*Sqrt[c*x^
2])/(d^3*(3 + m)*x) + (b^2*(d*x)^(4 + m)*Sqrt[c*x^2])/(d^4*(4 + m)*x)

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Rubi in Sympy [A]  time = 29.185, size = 82, normalized size = 0.87 \[ \frac{a^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 2}}{d^{2} x \left (m + 2\right )} + \frac{2 a b \sqrt{c x^{2}} \left (d x\right )^{m + 3}}{d^{3} x \left (m + 3\right )} + \frac{b^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 4}}{d^{4} x \left (m + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c*x**2)*(d*x)**(m + 2)/(d**2*x*(m + 2)) + 2*a*b*sqrt(c*x**2)*(d*x)**(m
+ 3)/(d**3*x*(m + 3)) + b**2*sqrt(c*x**2)*(d*x)**(m + 4)/(d**4*x*(m + 4))

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Mathematica [A]  time = 0.0557398, size = 48, normalized size = 0.51 \[ x \sqrt{c x^2} (d x)^m \left (\frac{a^2}{m+2}+\frac{2 a b x}{m+3}+\frac{b^2 x^2}{m+4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x*(d*x)^m*Sqrt[c*x^2]*(a^2/(2 + m) + (2*a*b*x)/(3 + m) + (b^2*x^2)/(4 + m))

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Maple [A]  time = 0.007, size = 95, normalized size = 1. \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+5\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+12\,abmx+6\,{b}^{2}{x}^{2}+7\,{a}^{2}m+16\,abx+12\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) }\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+5*b^2*m*x^2+a^2*m^2+12*a*b*m*x+6*b^2*x^2+7*a^2*m+16*a
*b*x+12*a^2)*(d*x)^m*(c*x^2)^(1/2)/(4+m)/(3+m)/(2+m)

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Maxima [A]  time = 1.37314, size = 86, normalized size = 0.91 \[ \frac{b^{2} \sqrt{c} d^{m} x^{4} x^{m}}{m + 4} + \frac{2 \, a b \sqrt{c} d^{m} x^{3} x^{m}}{m + 3} + \frac{a^{2} \sqrt{c} d^{m} x^{2} x^{m}}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*sqrt(c)*d^m*x^4*x^m/(m + 4) + 2*a*b*sqrt(c)*d^m*x^3*x^m/(m + 3) + a^2*sqrt(c
)*d^m*x^2*x^m/(m + 2)

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Fricas [A]  time = 0.218961, size = 127, normalized size = 1.35 \[ \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 6 \, b^{2}\right )} x^{3} + 2 \,{\left (a b m^{2} + 6 \, a b m + 8 \, a b\right )} x^{2} +{\left (a^{2} m^{2} + 7 \, a^{2} m + 12 \, a^{2}\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^2*m^2 + 5*b^2*m + 6*b^2)*x^3 + 2*(a*b*m^2 + 6*a*b*m + 8*a*b)*x^2 + (a^2*m^2
+ 7*a^2*m + 12*a^2)*x)*sqrt(c*x^2)*(d*x)^m/(m^3 + 9*m^2 + 26*m + 24)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError