3.9.85 \(\int \frac {x^6}{(c x^2)^{3/2} (a+b x)} \, dx\) [885]

Optimal. Leaf size=95 \[ \frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}}-\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}} \]

[Out]

a^2*x^2/b^3/c/(c*x^2)^(1/2)-1/2*a*x^3/b^2/c/(c*x^2)^(1/2)+1/3*x^4/b/c/(c*x^2)^(1/2)-a^3*x*ln(b*x+a)/b^4/c/(c*x
^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \begin {gather*} -\frac {a^3 x \log (a+b x)}{b^4 c \sqrt {c x^2}}+\frac {a^2 x^2}{b^3 c \sqrt {c x^2}}-\frac {a x^3}{2 b^2 c \sqrt {c x^2}}+\frac {x^4}{3 b c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

(a^2*x^2)/(b^3*c*Sqrt[c*x^2]) - (a*x^3)/(2*b^2*c*Sqrt[c*x^2]) + x^4/(3*b*c*Sqrt[c*x^2]) - (a^3*x*Log[a + b*x])
/(b^4*c*Sqrt[c*x^2])

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

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Mathematica [A]
time = 0.01, size = 53, normalized size = 0.56 \begin {gather*} \frac {x^3 \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

(x^3*(b*x*(6*a^2 - 3*a*b*x + 2*b^2*x^2) - 6*a^3*Log[a + b*x]))/(6*b^4*(c*x^2)^(3/2))

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^6/((c*x^2)^(3/2)*(a + b*x)),x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.13, size = 52, normalized size = 0.55

method	result	size
default	\(-\frac {x^{3} \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 \left (c \,x^{2}\right )^{\frac {3}{2}} b^{4}}\)	\(52\)
risch	\(\frac {x \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{c \sqrt {c \,x^{2}}\, b^{3}}-\frac {a^{3} x \ln \left (b x +a \right )}{b^{4} c \sqrt {c \,x^{2}}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(c*x^2)^(3/2)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

-1/6*x^3*(-2*b^3*x^3+3*a*b^2*x^2+6*a^3*ln(b*x+a)-6*a^2*b*x)/(c*x^2)^(3/2)/b^4

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Maxima [A]
time = 0.31, size = 162, normalized size = 1.71 \begin {gather*} \frac {x^{4}}{3 \, \sqrt {c x^{2}} b c} - \frac {a x^{3}}{2 \, \sqrt {c x^{2}} b^{2} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} b^{3} c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4} c^{\frac {3}{2}}} + \frac {29 \, a^{3} x}{6 \, \sqrt {c x^{2}} b^{4} c} - \frac {a^{3} \log \left (b x\right )}{b^{4} c^{\frac {3}{2}}} - \frac {2 \, a^{4}}{\sqrt {c x^{2}} b^{5} c} + \frac {2 \, a^{4}}{b^{5} c^{\frac {3}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2)^(3/2)/(b*x+a),x, algorithm="maxima")

[Out]

1/3*x^4/(sqrt(c*x^2)*b*c) - 1/2*a*x^3/(sqrt(c*x^2)*b^2*c) + a^2*x^2/(sqrt(c*x^2)*b^3*c) - (-1)^(2*a*c*x/b)*a^3
*log(-2*a*c*x/(b*abs(b*x + a)))/(b^4*c^(3/2)) + 29/6*a^3*x/(sqrt(c*x^2)*b^4*c) - a^3*log(b*x)/(b^4*c^(3/2)) -
2*a^4/(sqrt(c*x^2)*b^5*c) + 2*a^4/(b^5*c^(3/2)*x)

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Fricas [A]
time = 0.29, size = 54, normalized size = 0.57 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2)^(3/2)/(b*x+a),x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^3 - 3*a*b^2*x^2 + 6*a^2*b*x - 6*a^3*log(b*x + a))*sqrt(c*x^2)/(b^4*c^2*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(c*x**2)**(3/2)/(b*x+a),x)

[Out]

Integral(x**6/((c*x**2)**(3/2)*(a + b*x)), x)

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Giac [A]
time = 0.00, size = 88, normalized size = 0.93 \begin {gather*} \frac {\frac {\frac {1}{3} b^{2} x^{3} \mathrm {sign}\left (x\right )^{2}-\frac {1}{2} a b x^{2} \mathrm {sign}\left (x\right )^{2}+a^{2} x \mathrm {sign}\left (x\right )^{2}}{b^{3} \mathrm {sign}\left (x\right )^{3}}-\frac {a^{3} \ln \left |b x+a\right |}{b^{4} \mathrm {sign}\left (x\right )}+\frac {a^{3} \ln \left |a\right |\cdot \mathrm {sign}\left (x\right )}{b^{4}}}{\sqrt {c} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(c*x^2)^(3/2)/(b*x+a),x)

[Out]

1/6*(6*a^3*log(abs(a))*sgn(x)/b^4 - 6*a^3*log(abs(b*x + a))/(b^4*sgn(x)) + (2*b^2*x^3*sgn(x)^2 - 3*a*b*x^2*sgn
(x)^2 + 6*a^2*x*sgn(x)^2)/(b^3*sgn(x)^3))/c^(3/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^6}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((c*x^2)^(3/2)*(a + b*x)),x)

[Out]

int(x^6/((c*x^2)^(3/2)*(a + b*x)), x)

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