3.28.75 \(\int \frac {(c+(a x+\sqrt {-b+a^2 x^2})^{3/4})^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx\)

Optimal. Leaf size=265 \[ \frac {4 c^{4/3} \log \left (\sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}-\sqrt [3]{c}\right )}{3 a}-\frac {2 c^{4/3} \log \left (\sqrt [3]{c} \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}+\left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{2/3}+c^{2/3}\right )}{3 a}-\frac {4 c^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a}+\frac {4 c \sqrt [3]{\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c}}{a}+\frac {\left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{4/3}}{a} \]

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Rubi [F]  time = 0.40, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.36, size = 199, normalized size = 0.75 \begin {gather*} \frac {-2 c^2 \left (\frac {c \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt {a^2 x^2-b}+a x}{\sqrt {a^2 x^2-b}+a x}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};-\frac {c}{\left (a x+\sqrt {a^2 x^2-b}\right )^{3/4}}\right )+6 c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+\left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}+5 c^2}{a \left (\left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}+c\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c^2 + 6*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4) + (a*x + Sqrt[-b + a^2*x^2])^(3/2) - 2*c^2*((a*x + Sqrt[-b + a^2
*x^2] + c*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(a*x + Sqrt[-b + a^2*x^2]))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3,
-(c/(a*x + Sqrt[-b + a^2*x^2])^(3/4))])/(a*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(2/3))

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IntegrateAlgebraic [A]  time = 0.96, size = 288, normalized size = 1.09 \begin {gather*} \frac {5 c \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{a}+\frac {\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4} \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{a}-\frac {4 c^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a}+\frac {4 c^{4/3} \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}\right )}{3 a}-\frac {2 c^{4/3} \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}+\left (c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}\right )^{2/3}\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/a + ((a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b +
a^2*x^2])^(3/4))^(1/3))/a - (4*c^(4/3)*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/(S
qrt[3]*c^(1/3))])/(Sqrt[3]*a) + (4*c^(4/3)*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3)])/(3*a)
- (2*c^(4/3)*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x
^2])^(3/4))^(2/3)])/(3*a)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {3}{4}}\right )^{\frac {4}{3}}}{\sqrt {a^{2} x^{2}-b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}\right )}^{\frac {4}{3}}}{\sqrt {a^{2} x^{2} - b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c + (a*x + sqrt(a^2*x^2 - b))^(3/4))^(4/3)/sqrt(a^2*x^2 - b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{3/4}\right )}^{4/3}}{\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + \left (a x + \sqrt {a^{2} x^{2} - b}\right )^{\frac {3}{4}}\right )^{\frac {4}{3}}}{\sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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