3.9.87 \(\int \frac {x^4}{(c x^2)^{3/2} (a+b x)} \, dx\) [887]

Optimal. Leaf size=45 \[ \frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, 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 {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

x^2/(b*c*Sqrt[c*x^2]) - (a*x*Log[a + b*x])/(b^2*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^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {x}{a+b x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 29, normalized size = 0.64 \begin {gather*} \frac {x^3 (b x-a \log (a+b x))}{b^2 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

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Maple [A]
time = 0.14, size = 29, normalized size = 0.64

method	result	size
default	\(-\frac {x^{3} \left (a \ln \left (b x +a \right )-b x \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} b^{2}}\)	\(29\)
risch	\(\frac {x^{2}}{b c \sqrt {c \,x^{2}}}-\frac {a x \ln \left (b x +a \right )}{b^{2} c \sqrt {c \,x^{2}}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(c*x^2)^(3/2)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (41) = 82\).
time = 0.29, size = 116, normalized size = 2.58 \begin {gather*} \frac {x^{2}}{\sqrt {c x^{2}} b c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} c^{\frac {3}{2}}} + \frac {2 \, a x}{\sqrt {c x^{2}} b^{2} c} - \frac {a \log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} - \frac {2 \, a^{2}}{\sqrt {c x^{2}} b^{3} c} + \frac {2 \, a^{2}}{b^{3} c^{\frac {3}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.37, size = 30, normalized size = 0.67 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a),x, algorithm="fricas")

[Out]

sqrt(c*x^2)*(b*x - a*log(b*x + a))/(b^2*c^2*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\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**4/(c*x**2)**(3/2)/(b*x+a),x)

[Out]

Integral(x**4/((c*x**2)**(3/2)*(a + b*x)), x)

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Giac [A]
time = 1.21, size = 50, normalized size = 1.11 \begin {gather*} \frac {\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2} \sqrt {c}} + \frac {x}{b \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(c*x^2)^(3/2)/(b*x+a),x, algorithm="giac")

[Out]

(a*log(abs(a))*sgn(x)/(b^2*sqrt(c)) + x/(b*sqrt(c)*sgn(x)) - a*log(abs(b*x + a))/(b^2*sqrt(c)*sgn(x)))/c

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^4}{{\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^4/((c*x^2)^(3/2)*(a + b*x)),x)

[Out]

int(x^4/((c*x^2)^(3/2)*(a + b*x)), x)

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