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

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)*Sqrt[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 = 0.0489113, 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, Rules used = {15, 16, 43} \[ \frac{a^2 c^2 \sqrt{c x^2} (d x)^{m+6}}{d^6 (m+6) x}+\frac{2 a b c^2 \sqrt{c x^2} (d x)^{m+7}}{d^7 (m+7) x}+\frac{b^2 c^2 \sqrt{c x^2} (d x)^{m+8}}{d^8 (m+8) x} \]

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)*Sqrt[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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d x)^m \left (c x^2\right )^{5/2} (a+b x)^2 \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^5 (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int (d x)^{5+m} (a+b x)^2 \, dx}{d^5 x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a^2 (d x)^{5+m}+\frac{2 a b (d x)^{6+m}}{d}+\frac{b^2 (d x)^{7+m}}{d^2}\right ) \, dx}{d^5 x}\\ &=\frac{a^2 c^2 (d x)^{6+m} \sqrt{c x^2}}{d^6 (6+m) x}+\frac{2 a b c^2 (d x)^{7+m} \sqrt{c x^2}}{d^7 (7+m) x}+\frac{b^2 c^2 (d x)^{8+m} \sqrt{c x^2}}{d^8 (8+m) x}\\ \end{align*}

Mathematica [A]  time = 0.0703456, 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 = 0.005, size = 95, normalized size = 0.9 \begin{align*}{\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}}}} \end{align*}

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+96*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 = 1.07237, size = 86, normalized size = 0.83 \begin{align*} \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} \end{align*}

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, 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 = 1.37286, size = 265, normalized size = 2.57 \begin{align*} \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} \end{align*}

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, 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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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, algorithm="giac")

[Out]

Exception raised: TypeError