3.707 \(\int \frac{1}{(a+b x) \left (c+d x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=268 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )} \]

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a
^2*d)^(3/4)) - (Sqrt[b]*ArcTanh[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4
)])/(b^2*c + a^2*d)^(3/4) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[-((b*Sqrt[c
])/Sqrt[b^2*c + a^2*d]), ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d
)*x) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sqrt[b^2*c + a^2*d],
ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x)

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Rubi [A]  time = 0.732715, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{c+d x^2}}{\sqrt [4]{a^2 d+b^2 c}}\right )}{\left (a^2 d+b^2 c\right )^{3/4}}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (-\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )}+\frac{a \sqrt [4]{c} \sqrt{-\frac{d x^2}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{d a^2+b^2 c}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d x^2+c}}{\sqrt [4]{c}}\right )\right |-1\right )}{x \left (a^2 d+b^2 c\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)*(c + d*x^2)^(3/4)),x]

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4)])/(b^2*c + a
^2*d)^(3/4)) - (Sqrt[b]*ArcTanh[(Sqrt[b]*(c + d*x^2)^(1/4))/(b^2*c + a^2*d)^(1/4
)])/(b^2*c + a^2*d)^(3/4) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[-((b*Sqrt[c
])/Sqrt[b^2*c + a^2*d]), ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d
)*x) + (a*c^(1/4)*Sqrt[-((d*x^2)/c)]*EllipticPi[(b*Sqrt[c])/Sqrt[b^2*c + a^2*d],
ArcSin[(c + d*x^2)^(1/4)/c^(1/4)], -1])/((b^2*c + a^2*d)*x)

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Rubi in Sympy [A]  time = 105.259, size = 238, normalized size = 0.89 \[ \frac{a \sqrt [4]{c} \sqrt{- \frac{d x^{2}}{c}} \Pi \left (- \frac{b \sqrt{c}}{\sqrt{a^{2} d + b^{2} c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{c + d x^{2}}}{\sqrt [4]{c}} \right )}\middle | -1\right )}{x \left (a^{2} d + b^{2} c\right )} + \frac{a \sqrt [4]{c} \sqrt{- \frac{d x^{2}}{c}} \Pi \left (\frac{b \sqrt{c}}{\sqrt{a^{2} d + b^{2} c}}; \operatorname{asin}{\left (\frac{\sqrt [4]{c + d x^{2}}}{\sqrt [4]{c}} \right )}\middle | -1\right )}{x \left (a^{2} d + b^{2} c\right )} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt [4]{c + d x^{2}}}{\sqrt [4]{a^{2} d + b^{2} c}} \right )}}{\left (a^{2} d + b^{2} c\right )^{\frac{3}{4}}} - \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [4]{c + d x^{2}}}{\sqrt [4]{a^{2} d + b^{2} c}} \right )}}{\left (a^{2} d + b^{2} c\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)/(d*x**2+c)**(3/4),x)

[Out]

a*c**(1/4)*sqrt(-d*x**2/c)*elliptic_pi(-b*sqrt(c)/sqrt(a**2*d + b**2*c), asin((c
+ d*x**2)**(1/4)/c**(1/4)), -1)/(x*(a**2*d + b**2*c)) + a*c**(1/4)*sqrt(-d*x**2
/c)*elliptic_pi(b*sqrt(c)/sqrt(a**2*d + b**2*c), asin((c + d*x**2)**(1/4)/c**(1/
4)), -1)/(x*(a**2*d + b**2*c)) - sqrt(b)*atan(sqrt(b)*(c + d*x**2)**(1/4)/(a**2*
d + b**2*c)**(1/4))/(a**2*d + b**2*c)**(3/4) - sqrt(b)*atanh(sqrt(b)*(c + d*x**2
)**(1/4)/(a**2*d + b**2*c)**(1/4))/(a**2*d + b**2*c)**(3/4)

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Mathematica [C]  time = 0.141376, size = 128, normalized size = 0.48 \[ -\frac{2 \left (\frac{b \left (x-\sqrt{-\frac{c}{d}}\right )}{a+b x}\right )^{3/4} \left (\frac{b \left (\sqrt{-\frac{c}{d}}+x\right )}{a+b x}\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},\frac{3}{4};\frac{5}{2};\frac{a-b \sqrt{-\frac{c}{d}}}{a+b x},\frac{a+b \sqrt{-\frac{c}{d}}}{a+b x}\right )}{3 b \left (c+d x^2\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((a + b*x)*(c + d*x^2)^(3/4)),x]

[Out]

(-2*((b*(-Sqrt[-(c/d)] + x))/(a + b*x))^(3/4)*((b*(Sqrt[-(c/d)] + x))/(a + b*x))
^(3/4)*AppellF1[3/2, 3/4, 3/4, 5/2, (a - b*Sqrt[-(c/d)])/(a + b*x), (a + b*Sqrt[
-(c/d)])/(a + b*x)])/(3*b*(c + d*x^2)^(3/4))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{bx+a} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}^{\frac{3}{4}}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \left (c + d x^{2}\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{2} + c\right )}^{\frac{3}{4}}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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