3.842 \(\int \frac{(e x)^m (a+b x^4)}{\sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=123 \[ \frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)}-\frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} (b c (m+1)-a d (m+3)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{d e (m+1) (m+3) \sqrt{c+d x^4}} \]

[Out]

(b*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d*e*(3 + m)) - ((b*c*(1 + m) - a*d*(3 + m))*(e*x)^(1 + m)*Sqrt[1 + (d*x^4)/
c]*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)])/(d*e*(1 + m)*(3 + m)*Sqrt[c + d*x^4])

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Rubi [A]  time = 0.0590184, antiderivative size = 115, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {459, 365, 364} \[ \frac{\sqrt{\frac{d x^4}{c}+1} (e x)^{m+1} \left (\frac{a}{m+1}-\frac{b c}{d (m+3)}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )}{e \sqrt{c+d x^4}}+\frac{b \sqrt{c+d x^4} (e x)^{m+1}}{d e (m+3)} \]

Antiderivative was successfully verified.

[In]

Int[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]

[Out]

(b*(e*x)^(1 + m)*Sqrt[c + d*x^4])/(d*e*(3 + m)) + ((a/(1 + m) - (b*c)/(d*(3 + m)))*(e*x)^(1 + m)*Sqrt[1 + (d*x
^4)/c]*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)])/(e*Sqrt[c + d*x^4])

Rule 459

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(e x)^m \left (a+b x^4\right )}{\sqrt{c+d x^4}} \, dx &=\frac{b (e x)^{1+m} \sqrt{c+d x^4}}{d e (3+m)}-\left (-a+\frac{b c (1+m)}{d (3+m)}\right ) \int \frac{(e x)^m}{\sqrt{c+d x^4}} \, dx\\ &=\frac{b (e x)^{1+m} \sqrt{c+d x^4}}{d e (3+m)}-\frac{\left (\left (-a+\frac{b c (1+m)}{d (3+m)}\right ) \sqrt{1+\frac{d x^4}{c}}\right ) \int \frac{(e x)^m}{\sqrt{1+\frac{d x^4}{c}}} \, dx}{\sqrt{c+d x^4}}\\ &=\frac{b (e x)^{1+m} \sqrt{c+d x^4}}{d e (3+m)}+\frac{\left (a-\frac{b c (1+m)}{d (3+m)}\right ) (e x)^{1+m} \sqrt{1+\frac{d x^4}{c}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{4};\frac{5+m}{4};-\frac{d x^4}{c}\right )}{e (1+m) \sqrt{c+d x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0849124, size = 110, normalized size = 0.89 \[ \frac{x \sqrt{\frac{d x^4}{c}+1} (e x)^m \left (a (m+5) \, _2F_1\left (\frac{1}{2},\frac{m+1}{4};\frac{m+5}{4};-\frac{d x^4}{c}\right )+b (m+1) x^4 \, _2F_1\left (\frac{1}{2},\frac{m+5}{4};\frac{m+9}{4};-\frac{d x^4}{c}\right )\right )}{(m+1) (m+5) \sqrt{c+d x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*x)^m*(a + b*x^4))/Sqrt[c + d*x^4],x]

[Out]

(x*(e*x)^m*Sqrt[1 + (d*x^4)/c]*(a*(5 + m)*Hypergeometric2F1[1/2, (1 + m)/4, (5 + m)/4, -((d*x^4)/c)] + b*(1 +
m)*x^4*Hypergeometric2F1[1/2, (5 + m)/4, (9 + m)/4, -((d*x^4)/c)]))/((1 + m)*(5 + m)*Sqrt[c + d*x^4])

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( b{x}^{4}+a \right ){\frac{1}{\sqrt{d{x}^{4}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

int((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="fricas")

[Out]

integral((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c), x)

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Sympy [C]  time = 6.7225, size = 119, normalized size = 0.97 \begin{align*} \frac{a e^{m} x x^{m} \Gamma \left (\frac{m}{4} + \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right )} + \frac{b e^{m} x^{5} x^{m} \Gamma \left (\frac{m}{4} + \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{4} + \frac{5}{4} \\ \frac{m}{4} + \frac{9}{4} \end{matrix}\middle |{\frac{d x^{4} e^{i \pi }}{c}} \right )}}{4 \sqrt{c} \Gamma \left (\frac{m}{4} + \frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

a*e**m*x*x**m*gamma(m/4 + 1/4)*hyper((1/2, m/4 + 1/4), (m/4 + 5/4,), d*x**4*exp_polar(I*pi)/c)/(4*sqrt(c)*gamm
a(m/4 + 5/4)) + b*e**m*x**5*x**m*gamma(m/4 + 5/4)*hyper((1/2, m/4 + 5/4), (m/4 + 9/4,), d*x**4*exp_polar(I*pi)
/c)/(4*sqrt(c)*gamma(m/4 + 9/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )} \left (e x\right )^{m}}{\sqrt{d x^{4} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^4+a)/(d*x^4+c)^(1/2),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)*(e*x)^m/sqrt(d*x^4 + c), x)