3.975 \(\int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx\)

Optimal. Leaf size=103 \[ \frac{a^2 c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{2 a b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac{b^2 c^2 \sqrt{c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]

[Out]

(a^2*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (2*a*b*c^2*(d*x)^(7 + m)*S
qrt[c*x^2])/(d^7*(7 + m)*x) + (b^2*c^2*(d*x)^(8 + m)*Sqrt[c*x^2])/(d^8*(8 + m)*x
)

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Rubi [A]  time = 0.120371, antiderivative size = 103, 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 \[ \frac{a^2 c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{2 a b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac{b^2 c^2 \sqrt{c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(c*x^2)^(5/2)*(a + b*x)^2,x]

[Out]

(a^2*c^2*(d*x)^(6 + m)*Sqrt[c*x^2])/(d^6*(6 + m)*x) + (2*a*b*c^2*(d*x)^(7 + m)*S
qrt[c*x^2])/(d^7*(7 + m)*x) + (b^2*c^2*(d*x)^(8 + m)*Sqrt[c*x^2])/(d^8*(8 + m)*x
)

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Rubi in Sympy [A]  time = 31.6366, size = 92, normalized size = 0.89 \[ \frac{a^{2} c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 6}}{d^{6} x \left (m + 6\right )} + \frac{2 a b c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 7}}{d^{7} x \left (m + 7\right )} + \frac{b^{2} c^{2} \sqrt{c x^{2}} \left (d x\right )^{m + 8}}{d^{8} x \left (m + 8\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**2)**(5/2)*(b*x+a)**2,x)

[Out]

a**2*c**2*sqrt(c*x**2)*(d*x)**(m + 6)/(d**6*x*(m + 6)) + 2*a*b*c**2*sqrt(c*x**2)
*(d*x)**(m + 7)/(d**7*x*(m + 7)) + b**2*c**2*sqrt(c*x**2)*(d*x)**(m + 8)/(d**8*x
*(m + 8))

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Mathematica [A]  time = 0.0695777, size = 48, normalized size = 0.47 \[ x \left (c x^2\right )^{5/2} (d x)^m \left (\frac{a^2}{m+6}+\frac{2 a b x}{m+7}+\frac{b^2 x^2}{m+8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(c*x^2)^(5/2)*(a + b*x)^2,x]

[Out]

x*(d*x)^m*(c*x^2)^(5/2)*(a^2/(6 + m) + (2*a*b*x)/(7 + m) + (b^2*x^2)/(8 + m))

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Maple [A]  time = 0.007, size = 95, normalized size = 0.9 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,ab{m}^{2}x+13\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}+28\,abmx+42\,{b}^{2}{x}^{2}+15\,{a}^{2}m+96\,abx+56\,{a}^{2} \right ) x \left ( dx \right ) ^{m}}{ \left ( 8+m \right ) \left ( 7+m \right ) \left ( 6+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2)^(5/2)*(b*x+a)^2,x)

[Out]

x*(b^2*m^2*x^2+2*a*b*m^2*x+13*b^2*m*x^2+a^2*m^2+28*a*b*m*x+42*b^2*x^2+15*a^2*m+9
6*a*b*x+56*a^2)*(d*x)^m*(c*x^2)^(5/2)/(8+m)/(7+m)/(6+m)

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Maxima [A]  time = 1.36166, size = 86, normalized size = 0.83 \[ \frac{b^{2} c^{\frac{5}{2}} d^{m} x^{8} x^{m}}{m + 8} + \frac{2 \, a b c^{\frac{5}{2}} d^{m} x^{7} x^{m}}{m + 7} + \frac{a^{2} c^{\frac{5}{2}} d^{m} x^{6} x^{m}}{m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^2*(d*x)^m,x, algorithm="maxima")

[Out]

b^2*c^(5/2)*d^m*x^8*x^m/(m + 8) + 2*a*b*c^(5/2)*d^m*x^7*x^m/(m + 7) + a^2*c^(5/2
)*d^m*x^6*x^m/(m + 6)

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Fricas [A]  time = 0.230234, size = 166, normalized size = 1.61 \[ \frac{{\left ({\left (b^{2} c^{2} m^{2} + 13 \, b^{2} c^{2} m + 42 \, b^{2} c^{2}\right )} x^{7} + 2 \,{\left (a b c^{2} m^{2} + 14 \, a b c^{2} m + 48 \, a b c^{2}\right )} x^{6} +{\left (a^{2} c^{2} m^{2} + 15 \, a^{2} c^{2} m + 56 \, a^{2} c^{2}\right )} x^{5}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{3} + 21 \, m^{2} + 146 \, m + 336} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)^2*(d*x)^m,x, algorithm="fricas")

[Out]

((b^2*c^2*m^2 + 13*b^2*c^2*m + 42*b^2*c^2)*x^7 + 2*(a*b*c^2*m^2 + 14*a*b*c^2*m +
48*a*b*c^2)*x^6 + (a^2*c^2*m^2 + 15*a^2*c^2*m + 56*a^2*c^2)*x^5)*sqrt(c*x^2)*(d
*x)^m/(m^3 + 21*m^2 + 146*m + 336)

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

[Out]

Timed out

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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)^(5/2)*(b*x + a)^2*(d*x)^m,x, algorithm="giac")

[Out]

Exception raised: TypeError