3.510 \(\int \frac {x^4}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\)

Optimal. Leaf size=88 \[ \frac {x^5 \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 \sqrt {a+b x^3} \sqrt {c+d x^3}} \]

[Out]

1/5*x^5*AppellF1(5/3,1/2,1/2,8/3,-b*x^3/a,-d*x^3/c)*(1+b*x^3/a)^(1/2)*(1+d*x^3/c)^(1/2)/(b*x^3+a)^(1/2)/(d*x^3
+c)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {511, 510} \[ \frac {x^5 \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 \sqrt {a+b x^3} \sqrt {c+d x^3}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(x^5*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*Sqrt
[a + b*x^3]*Sqrt[c + d*x^3])

Rule 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx &=\frac {\sqrt {1+\frac {b x^3}{a}} \int \frac {x^4}{\sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3}} \, dx}{\sqrt {a+b x^3}}\\ &=\frac {\left (\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}\right ) \int \frac {x^4}{\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}}} \, dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\\ &=\frac {x^5 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 \sqrt {a+b x^3} \sqrt {c+d x^3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 90, normalized size = 1.02 \[ \frac {x^5 \sqrt {\frac {a+b x^3}{a}} \sqrt {\frac {c+d x^3}{c}} F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{5 \sqrt {a+b x^3} \sqrt {c+d x^3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^5*Sqrt[(a + b*x^3)/a]*Sqrt[(c + d*x^3)/c]*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(5*Sqrt
[a + b*x^3]*Sqrt[c + d*x^3])

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{4}}{b d x^{6} + {\left (b c + a d\right )} x^{3} + a c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^4/(b*d*x^6 + (b*c + a*d)*x^3 + a*c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)

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maple [F]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

[Out]

int(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)),x)

[Out]

int(x^4/((a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

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

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