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

Optimal. Leaf size=61 \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]

[Out]

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

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Rubi [A]  time = 0.0674009, antiderivative size = 61, 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 \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5407, size = 53, normalized size = 0.87 \[ \frac{a c \sqrt{c x^{2}} \left (d x\right )^{m + 4}}{d^{4} x \left (m + 4\right )} + \frac{b c \sqrt{c x^{2}} \left (d x\right )^{m + 5}}{d^{5} x \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*sqrt(c*x**2)*(d*x)**(m + 4)/(d**4*x*(m + 4)) + b*c*sqrt(c*x**2)*(d*x)**(m +
5)/(d**5*x*(m + 5))

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Mathematica [A]  time = 0.0301478, size = 39, normalized size = 0.64 \[ \frac{\left (c x^2\right )^{3/2} (d x)^m \left (\frac{a x^4}{m+4}+\frac{b x^5}{m+5}\right )}{x^3} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^m*(c*x^2)^(3/2)*((a*x^4)/(4 + m) + (b*x^5)/(5 + m)))/x^3

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Maple [A]  time = 0.003, size = 40, normalized size = 0.7 \[{\frac{ \left ( bmx+am+4\,bx+5\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(b*m*x+a*m+4*b*x+5*a)*(d*x)^m*(c*x^2)^(3/2)/(5+m)/(4+m)

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Maxima [A]  time = 1.37678, size = 53, normalized size = 0.87 \[ \frac{b c^{\frac{3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac{a c^{\frac{3}{2}} d^{m} x^{4} x^{m}}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*c^(3/2)*d^m*x^5*x^m/(m + 5) + a*c^(3/2)*d^m*x^4*x^m/(m + 4)

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Fricas [A]  time = 0.222854, size = 68, normalized size = 1.11 \[ \frac{{\left ({\left (b c m + 4 \, b c\right )} x^{4} +{\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*c*m + 4*b*c)*x^4 + (a*c*m + 5*a*c)*x^3)*sqrt(c*x^2)*(d*x)^m/(m^2 + 9*m + 20)

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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)**(3/2)*(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((c*x^2)^(3/2)*(b*x + a)*(d*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError