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

Optimal. Leaf size=59 \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]

[Out]

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

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Rubi [A]  time = 0.0606329, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.6165, size = 49, normalized size = 0.83 \[ \frac{a \sqrt{c x^{2}} \left (d x\right )^{m + 2}}{d^{2} x \left (m + 2\right )} + \frac{b \sqrt{c x^{2}} \left (d x\right )^{m + 3}}{d^{3} x \left (m + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0256086, size = 39, normalized size = 0.66 \[ \frac{\sqrt{c x^2} (d x)^m \left (\frac{a x^2}{m+2}+\frac{b x^3}{m+3}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 0.7 \[{\frac{ \left ( bmx+am+2\,bx+3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 2+m \right ) }\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(b*m*x+a*m+2*b*x+3*a)*(d*x)^m*(c*x^2)^(1/2)/(3+m)/(2+m)

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Maxima [A]  time = 1.37139, size = 53, normalized size = 0.9 \[ \frac{b \sqrt{c} d^{m} x^{3} x^{m}}{m + 3} + \frac{a \sqrt{c} d^{m} x^{2} x^{m}}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232352, size = 59, normalized size = 1. \[ \frac{{\left ({\left (b m + 2 \, b\right )} x^{2} +{\left (a m + 3 \, a\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*m + 2*b)*x^2 + (a*m + 3*a)*x)*sqrt(c*x^2)*(d*x)^m/(m^2 + 5*m + 6)

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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),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)*(d*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError