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

Optimal. Leaf size=206 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{8 a^2 c^2 x}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2} \]

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(2*a*c*x^2) + ((7*b*c + 5*a*d)*(a + b*x)^(1/4
)*(c + d*x)^(3/4))/(8*a^2*c^2*x) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[
(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(11/4)*c^(9/4)) - ((
21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(
c + d*x)^(1/4))])/(16*a^(11/4)*c^(9/4))

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Rubi [A]  time = 0.37155, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (5 a d+7 b c)}{8 a^2 c^2 x}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt [4]{a+b x}}{\sqrt [4]{a} \sqrt [4]{c+d x}}\right )}{16 a^{11/4} c^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4}}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a + b*x)^(1/4)*(c + d*x)^(3/4))/(2*a*c*x^2) + ((7*b*c + 5*a*d)*(a + b*x)^(1/4
)*(c + d*x)^(3/4))/(8*a^2*c^2*x) - ((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTan[
(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(c + d*x)^(1/4))])/(16*a^(11/4)*c^(9/4)) - ((
21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(c^(1/4)*(a + b*x)^(1/4))/(a^(1/4)*(
c + d*x)^(1/4))])/(16*a^(11/4)*c^(9/4))

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Rubi in Sympy [A]  time = 45.6989, size = 194, normalized size = 0.94 \[ - \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}}}{2 a c x^{2}} + \frac{\sqrt [4]{a + b x} \left (c + d x\right )^{\frac{3}{4}} \left (5 a d + 7 b c\right )}{8 a^{2} c^{2} x} - \frac{\left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{16 a^{\frac{11}{4}} c^{\frac{9}{4}}} - \frac{\left (5 a^{2} d^{2} + 6 a b c d + 21 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{c} \sqrt [4]{a + b x}}{\sqrt [4]{a} \sqrt [4]{c + d x}} \right )}}{16 a^{\frac{11}{4}} c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(1/4)*(c + d*x)**(3/4)/(2*a*c*x**2) + (a + b*x)**(1/4)*(c + d*x)**(3
/4)*(5*a*d + 7*b*c)/(8*a**2*c**2*x) - (5*a**2*d**2 + 6*a*b*c*d + 21*b**2*c**2)*a
tan(c**(1/4)*(a + b*x)**(1/4)/(a**(1/4)*(c + d*x)**(1/4)))/(16*a**(11/4)*c**(9/4
)) - (5*a**2*d**2 + 6*a*b*c*d + 21*b**2*c**2)*atanh(c**(1/4)*(a + b*x)**(1/4)/(a
**(1/4)*(c + d*x)**(1/4)))/(16*a**(11/4)*c**(9/4))

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Mathematica [C]  time = 0.366507, size = 211, normalized size = 1.02 \[ \frac{\frac{2 b d x^3 \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )}{-8 b d x F_1\left (1;\frac{3}{4},\frac{1}{4};2;-\frac{a}{b x},-\frac{c}{d x}\right )+b c F_1\left (2;\frac{3}{4},\frac{5}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )+3 a d F_1\left (2;\frac{7}{4},\frac{1}{4};3;-\frac{a}{b x},-\frac{c}{d x}\right )}+(a+b x) (c+d x) (-4 a c+5 a d x+7 b c x)}{8 a^2 c^2 x^2 (a+b x)^{3/4} \sqrt [4]{c+d x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)*(c + d*x)*(-4*a*c + 7*b*c*x + 5*a*d*x) + (2*b*d*(21*b^2*c^2 + 6*a*b*c
*d + 5*a^2*d^2)*x^3*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))]
))/(8*a^2*c^2*x^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28084, size = 1597, normalized size = 7.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(4*a^2*c^2*x^2*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d
^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 1
5900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4)*arctan(
(a^3*c^2*d*x + a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c
^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5
+ 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8)/(a^11*c^9))^(1/4)/((2
1*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (d*x + c)*s
qrt(((441*b^4*c^4 + 252*a*b^3*c^3*d + 246*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 + 25*
a^4*d^4)*sqrt(b*x + a)*sqrt(d*x + c) + (a^6*c^4*d*x + a^6*c^5)*sqrt((194481*b^8*
c^8 + 222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112
806*a^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b
*c*d^7 + 625*a^8*d^8)/(a^11*c^9)))/(d*x + c)))) - a^2*c^2*x^2*((194481*b^8*c^8 +
222264*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a
^4*b^4*c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^
7 + 625*a^8*d^8)/(a^11*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*
x + a)^(1/4)*(d*x + c)^(3/4) + (a^3*c^2*d*x + a^3*c^3)*((194481*b^8*c^8 + 222264
*a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*
c^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625
*a^8*d^8)/(a^11*c^9))^(1/4))/(d*x + c)) + a^2*c^2*x^2*((194481*b^8*c^8 + 222264*
a*b^7*c^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c
^4*d^4 + 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*
a^8*d^8)/(a^11*c^9))^(1/4)*log(((21*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(b*x + a)^(
1/4)*(d*x + c)^(3/4) - (a^3*c^2*d*x + a^3*c^3)*((194481*b^8*c^8 + 222264*a*b^7*c
^7*d + 280476*a^2*b^6*c^6*d^2 + 176904*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4
+ 42120*a^5*b^3*c^3*d^5 + 15900*a^6*b^2*c^2*d^6 + 3000*a^7*b*c*d^7 + 625*a^8*d^8
)/(a^11*c^9))^(1/4))/(d*x + c)) - 4*(4*a*c - (7*b*c + 5*a*d)*x)*(b*x + a)^(1/4)*
(d*x + c)^(3/4))/(a^2*c^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*(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^3),x, algorithm="giac")

[Out]

Timed out