3.1706 \(\int \frac{(a+b x)^{7/4}}{(c+d x)^{3/4}} \, dx\)

Optimal. Leaf size=167 \[ -\frac{21 (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac{21 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}-\frac{7 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 d^2}+\frac{(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.21508, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{21 (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}+\frac{21 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 \sqrt [4]{b} d^{11/4}}-\frac{7 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)}{8 d^2}+\frac{(a+b x)^{7/4} \sqrt [4]{c+d x}}{2 d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(7/4)/(c + d*x)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.3413, size = 151, normalized size = 0.9 \[ \frac{\left (a + b x\right )^{\frac{7}{4}} \sqrt [4]{c + d x}}{2 d} + \frac{7 \left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x} \left (a d - b c\right )}{8 d^{2}} + \frac{21 \left (a d - b c\right )^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 \sqrt [4]{b} d^{\frac{11}{4}}} + \frac{21 \left (a d - b c\right )^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt [4]{c + d x}}{\sqrt [4]{d} \sqrt [4]{a + b x}} \right )}}{16 \sqrt [4]{b} d^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(7/4)/(d*x+c)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.196304, size = 107, normalized size = 0.64 \[ \frac{\sqrt [4]{c+d x} \left (21 (b c-a d)^2 \sqrt [4]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{b (c+d x)}{b c-a d}\right )+d (a+b x) (11 a d-7 b c+4 b d x)\right )}{8 d^3 \sqrt [4]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(7/4)/(c + d*x)^(3/4),x]

[Out]

((c + d*x)^(1/4)*(d*(a + b*x)*(-7*b*c + 11*a*d + 4*b*d*x) + 21*(b*c - a*d)^2*((d
*(a + b*x))/(-(b*c) + a*d))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (b*(c + d*x))
/(b*c - a*d)]))/(8*d^3*(a + b*x)^(1/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{1 \left ( bx+a \right ) ^{{\frac{7}{4}}} \left ( dx+c \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(7/4)/(d*x+c)^(3/4),x)

[Out]

int((b*x+a)^(7/4)/(d*x+c)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{\frac{7}{4}}}{{\left (d x + c\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(3/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(7/4)/(d*x + c)^(3/4), x)

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Fricas [A]  time = 0.254043, size = 1508, normalized size = 9.03 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(7/4)/(d*x + c)^(3/4),x, algorithm="fricas")

[Out]

-1/32*(84*d^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^
3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7
+ a^8*d^8)/(b*d^11))^(1/4)*arctan((b*d^3*x + a*d^3)*((b^8*c^8 - 8*a*b^7*c^7*d +
28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d
^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4)/((b^2*c^2 - 2
*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b*x + a)*sqrt(((b^4*c^4 -
4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*x + a)*sqrt
(d*x + c) + (b*d^6*x + a*d^6)*sqrt((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2
- 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2
*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11)))/(b*x + a)))) - 21*d^2*((b^8*c^8 - 8*a
*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a
^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4)*l
og(21*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b*d^3*
x + a*d^3)*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 +
70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 +
a^8*d^8)/(b*d^11))^(1/4))/(b*x + a)) + 21*d^2*((b^8*c^8 - 8*a*b^7*c^7*d + 28*a^2
*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4 - 56*a^5*b^3*c^3*d^5 + 28
*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^(1/4)*log(21*((b^2*c^2 - 2
*a*b*c*d + a^2*d^2)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (b*d^3*x + a*d^3)*((b^8*c^
8 - 8*a*b^7*c^7*d + 28*a^2*b^6*c^6*d^2 - 56*a^3*b^5*c^5*d^3 + 70*a^4*b^4*c^4*d^4
- 56*a^5*b^3*c^3*d^5 + 28*a^6*b^2*c^2*d^6 - 8*a^7*b*c*d^7 + a^8*d^8)/(b*d^11))^
(1/4))/(b*x + a)) - 4*(4*b*d*x - 7*b*c + 11*a*d)*(b*x + a)^(3/4)*(d*x + c)^(1/4)
)/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(7/4)/(d*x+c)**(3/4),x)

[Out]

Timed out

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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((b*x + a)^(7/4)/(d*x + c)^(3/4),x, algorithm="giac")

[Out]

Timed out