3.893 \(\int \frac{1}{x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=134 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) + ((3*b*c + a*d)*ArcTan[(c^(1/4)*(a
+ b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)*c^(5/4)) + ((3*b*c + a*d)*
ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)*c^(5/4)
)

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Rubi [A]  time = 0.196362, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(a d+3 b c) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}+\frac{(a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{2 a^{7/4} c^{5/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{a c x} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b*x)^(1/4)*(c + d*x)^(3/4))/(a*c*x)) + ((3*b*c + a*d)*ArcTan[(c^(1/4)*(a
+ b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)*c^(5/4)) + ((3*b*c + a*d)*
ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(2*a^(7/4)*c^(5/4)
)

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Rubi in Sympy [A]  time = 19.8673, size = 119, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{a c x} + \frac{\left (a d + 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{2 a^{\frac{7}{4}} c^{\frac{5}{4}}} + \frac{\left (a d + 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{2 a^{\frac{7}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(1/4)*(c + d*x)**(3/4)/(a*c*x) + (a*d + 3*b*c)*atan(c**(1/4)*(a + b*
x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(2*a**(7/4)*c**(5/4)) + (a*d + 3*b*c)*ata
nh(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(2*a**(7/4)*c**(5/4))

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Mathematica [C]  time = 0.3048, size = 180, normalized size = 1.34 \[ \frac{\frac{2 b d x^2 (a d+3 b c) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )-b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )-3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}-(a+b x) (c+d x)}{a c x (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-((a + b*x)*(c + d*x)) + (2*b*d*(3*b*c + a*d)*x^2*AppellF1[1, 3/4, 1/4, 2, -(a/
(b*x)), -(c/(d*x))])/(8*b*d*x*AppellF1[1, 3/4, 1/4, 2, -(a/(b*x)), -(c/(d*x))] -
b*c*AppellF1[2, 3/4, 5/4, 3, -(a/(b*x)), -(c/(d*x))] - 3*a*d*AppellF1[2, 7/4, 1
/4, 3, -(a/(b*x)), -(c/(d*x))]))/(a*c*x*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.258127, size = 903, normalized size = 6.74 \[ -\frac{4 \, a c x \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}}}{{\left (3 \, b c + a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (d x + c\right )} \sqrt{\frac{{\left (9 \, b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (a^{4} c^{2} d x + a^{4} c^{3}\right )} \sqrt{\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}}}{d x + c}}}\right ) - a c x \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (3 \, b c + a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} +{\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) + a c x \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (\frac{{\left (3 \, b c + a d\right )}{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}} -{\left (a^{2} c d x + a^{2} c^{2}\right )} \left (\frac{81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} c^{5}}\right )^{\frac{1}{4}}}{d x + c}\right ) + 4 \,{\left (b x + a\right )}^{\frac{1}{4}}{\left (d x + c\right )}^{\frac{3}{4}}}{4 \, a c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d
^3 + a^4*d^4)/(a^7*c^5))^(1/4)*arctan((a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a
*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)/((3
*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*sqrt(((9*b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (a^4*c^2*d*x + a^4*c^3)*sqrt((81*b
^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c
^5)))/(d*x + c)))) - a*c*x*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 +
12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(
d*x + c)^(3/4) + (a^2*c*d*x + a^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b
^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) + a*c*x*((81
*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7
*c^5))^(1/4)*log(((3*b*c + a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (a^2*c*d*x + a
^2*c^2)*((81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a
^4*d^4)/(a^7*c^5))^(1/4))/(d*x + c)) + 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(a*c*x
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out