3.9.14 \(\int x (c x^2)^{3/2} (a+b x)^2 \, dx\) [814]

Optimal. Leaf size=60 \[ \frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \begin {gather*} \frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(c*x^2)^(3/2)*(a + b*x)^2,x]

[Out]

(a^2*c*x^4*Sqrt[c*x^2])/5 + (a*b*c*x^5*Sqrt[c*x^2])/3 + (b^2*c*x^6*Sqrt[c*x^2])/7

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 45

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 x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x^4 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a^2 x^4+2 a b x^5+b^2 x^6\right ) \, dx}{x}\\ &=\frac {1}{5} a^2 c x^4 \sqrt {c x^2}+\frac {1}{3} a b c x^5 \sqrt {c x^2}+\frac {1}{7} b^2 c x^6 \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.58 \begin {gather*} \frac {1}{105} x^2 \left (c x^2\right )^{3/2} \left (21 a^2+35 a b x+15 b^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(c*x^2)^(3/2)*(21*a^2 + 35*a*b*x + 15*b^2*x^2))/105

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Mathics [A]
time = 1.93, size = 30, normalized size = 0.50 \begin {gather*} x^2 \left (\frac {a^2}{5}+\frac {a b x}{3}+\frac {b^2 x^2}{7}\right ) {\left (c x^2\right )}^{\frac {3}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*(c*x^2)^(3/2)*(a + b*x)^2,x]')

[Out]

x ^ 2 (a ^ 2 / 5 + a b x / 3 + b ^ 2 x ^ 2 / 7) (c x ^ 2) ^ (3 / 2)

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Maple [A]
time = 0.11, size = 32, normalized size = 0.53

method	result	size
gosper	\(\frac {x^{2} \left (15 x^{2} b^{2}+35 a b x +21 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{105}\)	\(32\)
default	\(\frac {x^{2} \left (15 x^{2} b^{2}+35 a b x +21 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{105}\)	\(32\)
risch	\(\frac {a^{2} c \,x^{4} \sqrt {c \,x^{2}}}{5}+\frac {a b c \,x^{5} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} c \,x^{6} \sqrt {c \,x^{2}}}{7}\)	\(49\)
trager	\(\frac {c \left (15 b^{2} x^{6}+35 a b \,x^{5}+15 b^{2} x^{5}+21 a^{2} x^{4}+35 a b \,x^{4}+15 b^{2} x^{4}+21 a^{2} x^{3}+35 a b \,x^{3}+15 b^{2} x^{3}+21 a^{2} x^{2}+35 a b \,x^{2}+15 x^{2} b^{2}+21 a^{2} x +35 a b x +15 b^{2} x +21 a^{2}+35 a b +15 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{105 x}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2)^(3/2)*(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/105*x^2*(15*b^2*x^2+35*a*b*x+21*a^2)*(c*x^2)^(3/2)

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Maxima [A]
time = 0.26, size = 49, normalized size = 0.82 \begin {gather*} \frac {\left (c x^{2}\right )^{\frac {5}{2}} b^{2} x^{2}}{7 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a b x}{3 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a^{2}}{5 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^(3/2)*(b*x+a)^2,x, algorithm="maxima")

[Out]

1/7*(c*x^2)^(5/2)*b^2*x^2/c + 1/3*(c*x^2)^(5/2)*a*b*x/c + 1/5*(c*x^2)^(5/2)*a^2/c

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Fricas [A]
time = 0.29, size = 36, normalized size = 0.60 \begin {gather*} \frac {1}{105} \, {\left (15 \, b^{2} c x^{6} + 35 \, a b c x^{5} + 21 \, a^{2} c x^{4}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^(3/2)*(b*x+a)^2,x, algorithm="fricas")

[Out]

1/105*(15*b^2*c*x^6 + 35*a*b*c*x^5 + 21*a^2*c*x^4)*sqrt(c*x^2)

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Sympy [A]
time = 0.23, size = 49, normalized size = 0.82 \begin {gather*} \frac {a^{2} x^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{5} + \frac {a b x^{3} \left (c x^{2}\right )^{\frac {3}{2}}}{3} + \frac {b^{2} x^{4} \left (c x^{2}\right )^{\frac {3}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x**2)**(3/2)*(b*x+a)**2,x)

[Out]

a**2*x**2*(c*x**2)**(3/2)/5 + a*b*x**3*(c*x**2)**(3/2)/3 + b**2*x**4*(c*x**2)**(3/2)/7

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Giac [A]
time = 0.00, size = 40, normalized size = 0.67 \begin {gather*} \sqrt {c} c \left (\frac {1}{5} a^{2} x^{5} \mathrm {sign}\left (x\right )+\frac {1}{7} b^{2} x^{7} \mathrm {sign}\left (x\right )+\frac {1}{3} a b x^{6} \mathrm {sign}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(c*x^2)^(3/2)*(b*x+a)^2,x)

[Out]

1/105*(15*b^2*x^7*sgn(x) + 35*a*b*x^6*sgn(x) + 21*a^2*x^5*sgn(x))*c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2)^(3/2)*(a + b*x)^2,x)

[Out]

int(x*(c*x^2)^(3/2)*(a + b*x)^2, x)

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