3.9.21 \(\int (c x^2)^{5/2} (a+b x)^2 \, dx\) [821]

Optimal. Leaf size=66 \[ \frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2} \]

[Out]

1/6*a^2*c^2*x^5*(c*x^2)^(1/2)+2/7*a*b*c^2*x^6*(c*x^2)^(1/2)+1/8*b^2*c^2*x^7*(c*x^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \begin {gather*} \frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(5/2)*(a + b*x)^2,x]

[Out]

(a^2*c^2*x^5*Sqrt[c*x^2])/6 + (2*a*b*c^2*x^6*Sqrt[c*x^2])/7 + (b^2*c^2*x^7*Sqrt[c*x^2])/8

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 \left (c x^2\right )^{5/2} (a+b x)^2 \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (a+b x)^2 \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx}{x}\\ &=\frac {1}{6} a^2 c^2 x^5 \sqrt {c x^2}+\frac {2}{7} a b c^2 x^6 \sqrt {c x^2}+\frac {1}{8} b^2 c^2 x^7 \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.50 \begin {gather*} \frac {1}{168} x \left (c x^2\right )^{5/2} \left (28 a^2+48 a b x+21 b^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c*x^2)^(5/2)*(28*a^2 + 48*a*b*x + 21*b^2*x^2))/168

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Maple [A]
time = 0.10, size = 30, normalized size = 0.45

method	result	size
gosper	\(\frac {x \left (21 x^{2} b^{2}+48 a b x +28 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{168}\)	\(30\)
default	\(\frac {x \left (21 x^{2} b^{2}+48 a b x +28 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{168}\)	\(30\)
risch	\(\frac {a^{2} c^{2} x^{5} \sqrt {c \,x^{2}}}{6}+\frac {2 a b \,c^{2} x^{6} \sqrt {c \,x^{2}}}{7}+\frac {b^{2} c^{2} x^{7} \sqrt {c \,x^{2}}}{8}\)	\(55\)
trager	\(\frac {c^{2} \left (21 b^{2} x^{7}+48 a b \,x^{6}+21 b^{2} x^{6}+28 a^{2} x^{5}+48 a b \,x^{5}+21 b^{2} x^{5}+28 a^{2} x^{4}+48 a b \,x^{4}+21 b^{2} x^{4}+28 a^{2} x^{3}+48 a b \,x^{3}+21 b^{2} x^{3}+28 a^{2} x^{2}+48 a b \,x^{2}+21 x^{2} b^{2}+28 a^{2} x +48 a b x +21 b^{2} x +28 a^{2}+48 a b +21 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{168 x}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/168*x*(21*b^2*x^2+48*a*b*x+28*a^2)*(c*x^2)^(5/2)

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Maxima [A]
time = 0.27, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, \left (c x^{2}\right )^{\frac {5}{2}} a^{2} x + \frac {\left (c x^{2}\right )^{\frac {7}{2}} b^{2} x}{8 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {7}{2}} a b}{7 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^2,x, algorithm="maxima")

[Out]

1/6*(c*x^2)^(5/2)*a^2*x + 1/8*(c*x^2)^(7/2)*b^2*x/c + 2/7*(c*x^2)^(7/2)*a*b/c

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Fricas [A]
time = 0.42, size = 42, normalized size = 0.64 \begin {gather*} \frac {1}{168} \, {\left (21 \, b^{2} c^{2} x^{7} + 48 \, a b c^{2} x^{6} + 28 \, a^{2} c^{2} x^{5}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^2,x, algorithm="fricas")

[Out]

1/168*(21*b^2*c^2*x^7 + 48*a*b*c^2*x^6 + 28*a^2*c^2*x^5)*sqrt(c*x^2)

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Sympy [A]
time = 0.30, size = 49, normalized size = 0.74 \begin {gather*} \frac {a^{2} x \left (c x^{2}\right )^{\frac {5}{2}}}{6} + \frac {2 a b x^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{7} + \frac {b^{2} x^{3} \left (c x^{2}\right )^{\frac {5}{2}}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x*(c*x**2)**(5/2)/6 + 2*a*b*x**2*(c*x**2)**(5/2)/7 + b**2*x**3*(c*x**2)**(5/2)/8

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Giac [A]
time = 1.81, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{168} \, {\left (21 \, b^{2} c^{2} x^{8} \mathrm {sgn}\left (x\right ) + 48 \, a b c^{2} x^{7} \mathrm {sgn}\left (x\right ) + 28 \, a^{2} c^{2} x^{6} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/168*(21*b^2*c^2*x^8*sgn(x) + 48*a*b*c^2*x^7*sgn(x) + 28*a^2*c^2*x^6*sgn(x))*sqrt(c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(5/2)*(a + b*x)^2,x)

[Out]

int((c*x^2)^(5/2)*(a + b*x)^2, x)

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