3.6.80 \(\int \frac {(a+b x^n+c x^{2 n})^{3/2}}{x^2} \, dx\) [580]

Optimal. Leaf size=150 \[ -\frac {a \sqrt {a+b x^n+c x^{2 n}} F_1\left (-\frac {1}{n};-\frac {3}{2},-\frac {3}{2};-\frac {1-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \]

[Out]

-a*AppellF1(-1/n,-3/2,-3/2,(-1+n)/n,-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*(a+b*x^n+
c*x^(2*n))^(1/2)/x/(1+2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(1+2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1399, 524} \begin {gather*} -\frac {a \sqrt {a+b x^n+c x^{2 n}} F_1\left (-\frac {1}{n};-\frac {3}{2},-\frac {3}{2};-\frac {1-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Sqrt[a + b*x^n + c*x^(2*n)]*AppellF1[-n^(-1), -3/2, -3/2, -((1 - n)/n), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]
), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/(x*Sqrt[1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^n)/(b +
Sqrt[b^2 - 4*a*c])]))

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1399

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a +
b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2*c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^
2 - 4*a*c, 2])))^FracPart[p])), Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2*c*(x^n/(b - Sqrt[
b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^n+c x^{2 n}\right )^{3/2}}{x^2} \, dx &=\frac {\left (a \sqrt {a+b x^n+c x^{2 n}}\right ) \int \frac {\left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{3/2}}{x^2} \, dx}{\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}\\ &=-\frac {a \sqrt {a+b x^n+c x^{2 n}} F_1\left (-\frac {1}{n};-\frac {3}{2},-\frac {3}{2};-\frac {1-n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(526\) vs. \(2(150)=300\).
time = 1.01, size = 526, normalized size = 3.51 \begin {gather*} \frac {2 (-1+n) \left (4 a^2 c \left (1-6 n+8 n^2\right )+x^n \left (b+c x^n\right ) \left (3 b^2 n^2+2 b c \left (2-9 n+7 n^2\right ) x^n+4 c^2 \left (1-3 n+2 n^2\right ) x^{2 n}\right )+a \left (3 b^2 n^2+2 b c \left (4-21 n+23 n^2\right ) x^n+4 c^2 \left (2-9 n+10 n^2\right ) x^{2 n}\right )\right )-6 a (-1+n) n^2 \left (b^2+4 a c (-1+2 n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} F_1\left (-\frac {1}{n};\frac {1}{2},\frac {1}{2};\frac {-1+n}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-3 b \left (4 a c (2-3 n)+b^2 (-2+n)\right ) n^2 x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {-1+n}{n};\frac {1}{2},\frac {1}{2};2-\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{8 c (-1+n)^2 (-1+2 n) (-1+3 n) x \sqrt {a+x^n \left (b+c x^n\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1 + n)*(4*a^2*c*(1 - 6*n + 8*n^2) + x^n*(b + c*x^n)*(3*b^2*n^2 + 2*b*c*(2 - 9*n + 7*n^2)*x^n + 4*c^2*(1 -
3*n + 2*n^2)*x^(2*n)) + a*(3*b^2*n^2 + 2*b*c*(4 - 21*n + 23*n^2)*x^n + 4*c^2*(2 - 9*n + 10*n^2)*x^(2*n))) - 6
*a*(-1 + n)*n^2*(b^2 + 4*a*c*(-1 + 2*n))*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[
(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])]*AppellF1[-n^(-1), 1/2, 1/2, (-1 + n)/n, (-2*c*x^n)/
(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[b^2 - 4*a*c])] - 3*b*(4*a*c*(2 - 3*n) + b^2*(-2 + n))*n^2*x^n*Sq
rt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt
[b^2 - 4*a*c])]*AppellF1[(-1 + n)/n, 1/2, 1/2, 2 - n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b +
Sqrt[b^2 - 4*a*c])])/(8*c*(-1 + n)^2*(-1 + 2*n)*(-1 + 3*n)*x*Sqrt[a + x^n*(b + c*x^n)])

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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