3.28.75 \(\int \frac {(c+(a x+\sqrt {-b+a^2 x^2})^{3/4})^{4/3}}{\sqrt {-b+a^2 x^2}} \, dx\) [2775]

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

[Out]

4*c*(c+(a*x+(a^2*x^2-b)^(1/2))^(3/4))^(1/3)/a+(c+(a*x+(a^2*x^2-b)^(1/2))^(3/4))^(4/3)/a-4/3*c^(4/3)*arctan(1/3
*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(3/4))^(1/3)*3^(1/2)/c^(1/3))*3^(1/2)/a+4/3*c^(4/3)*ln(-c^(1/3)+(c+(a*
x+(a^2*x^2-b)^(1/2))^(3/4))^(1/3))/a-2/3*c^(4/3)*ln(c^(2/3)+c^(1/3)*(c+(a*x+(a^2*x^2-b)^(1/2))^(3/4))^(1/3)+(c
+(a*x+(a^2*x^2-b)^(1/2))^(3/4))^(2/3))/a

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Rubi [F]
time = 0.28, 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 [A]
time = 0.89, size = 274, normalized size = 1.03 \begin {gather*} \frac {15 c \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}+3 \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}}-4 \sqrt {3} c^{4/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{c+\left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}}{\sqrt [3]{c}}}{\sqrt {3}}\right )+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 )-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]

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

[Out]

(15*c*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3) + 3*(a*x + Sqrt[-b + a^2*x^2])^(3/4)*(c + (a*x + Sqrt[-b +
a^2*x^2])^(3/4))^(1/3) - 4*Sqrt[3]*c^(4/3)*ArcTan[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3))/c^(1/3
))/Sqrt[3]] + 4*c^(4/3)*Log[-c^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(3/4))^(1/3)] - 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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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*} \text {Failed to integrate} \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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Fricas [F(-1)] Timed out
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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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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Giac [F(-1)] Timed out
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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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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