3.1833 \(\int \frac{(c+d x)^{7/6}}{(a+b x)^{7/6}} \, dx\)

Optimal. Leaf size=403 \[ -\frac{7 \sqrt [6]{d} (b c-a d) \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt{3} b^{13/6}}-\frac{7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}+\frac{7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac{6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}} \]

[Out]

(7*d*(a + b*x)^(5/6)*(c + d*x)^(1/6))/b^2 - (6*(c + d*x)^(7/6))/(b*(a + b*x)^(1/
6)) + (7*d^(1/6)*(b*c - a*d)*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqr
t[3]*b^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(13/6)) - (7*d^(1/6)*(b*c - a*d)*Ar
cTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])
/(2*Sqrt[3]*b^(13/6)) + (7*d^(1/6)*(b*c - a*d)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))
/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(13/6)) - (7*d^(1/6)*(b*c - a*d)*Log[b^(1/3) +
(d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(
c + d*x)^(1/6)])/(12*b^(13/6)) + (7*d^(1/6)*(b*c - a*d)*Log[b^(1/3) + (d^(1/3)*(
a + b*x)^(1/3))/(c + d*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c + d*x)^(1
/6)])/(12*b^(13/6))

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Rubi [A]  time = 1.05817, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{7 \sqrt [6]{d} (b c-a d) \log \left (-\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \log \left (\frac{\sqrt [6]{b} \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{c+d x}}+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}+\sqrt [3]{b}\right )}{12 b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{2 \sqrt{3} b^{13/6}}-\frac{7 \sqrt [6]{d} (b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt{3} \sqrt [6]{b} \sqrt [6]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt{3} b^{13/6}}+\frac{7 \sqrt [6]{d} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt [6]{d} \sqrt [6]{a+b x}}{\sqrt [6]{b} \sqrt [6]{c+d x}}\right )}{3 b^{13/6}}+\frac{7 d (a+b x)^{5/6} \sqrt [6]{c+d x}}{b^2}-\frac{6 (c+d x)^{7/6}}{b \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(7/6)/(a + b*x)^(7/6),x]

[Out]

(7*d*(a + b*x)^(5/6)*(c + d*x)^(1/6))/b^2 - (6*(c + d*x)^(7/6))/(b*(a + b*x)^(1/
6)) + (7*d^(1/6)*(b*c - a*d)*ArcTan[1/Sqrt[3] - (2*d^(1/6)*(a + b*x)^(1/6))/(Sqr
t[3]*b^(1/6)*(c + d*x)^(1/6))])/(2*Sqrt[3]*b^(13/6)) - (7*d^(1/6)*(b*c - a*d)*Ar
cTan[1/Sqrt[3] + (2*d^(1/6)*(a + b*x)^(1/6))/(Sqrt[3]*b^(1/6)*(c + d*x)^(1/6))])
/(2*Sqrt[3]*b^(13/6)) + (7*d^(1/6)*(b*c - a*d)*ArcTanh[(d^(1/6)*(a + b*x)^(1/6))
/(b^(1/6)*(c + d*x)^(1/6))])/(3*b^(13/6)) - (7*d^(1/6)*(b*c - a*d)*Log[b^(1/3) +
(d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3) - (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(
c + d*x)^(1/6)])/(12*b^(13/6)) + (7*d^(1/6)*(b*c - a*d)*Log[b^(1/3) + (d^(1/3)*(
a + b*x)^(1/3))/(c + d*x)^(1/3) + (b^(1/6)*d^(1/6)*(a + b*x)^(1/6))/(c + d*x)^(1
/6)])/(12*b^(13/6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(7/6)/(b*x+a)**(7/6),x)

[Out]

Timed out

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Mathematica [C]  time = 0.205424, size = 93, normalized size = 0.23 \[ \frac{\sqrt [6]{c+d x} \left (7 (b c-a d) \sqrt [6]{\frac{d (a+b x)}{a d-b c}} \, _2F_1\left (\frac{1}{6},\frac{1}{6};\frac{7}{6};\frac{b (c+d x)}{b c-a d}\right )+7 a d-6 b c+b d x\right )}{b^2 \sqrt [6]{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(7/6)/(a + b*x)^(7/6),x]

[Out]

((c + d*x)^(1/6)*(-6*b*c + 7*a*d + b*d*x + 7*(b*c - a*d)*((d*(a + b*x))/(-(b*c)
+ a*d))^(1/6)*Hypergeometric2F1[1/6, 1/6, 7/6, (b*(c + d*x))/(b*c - a*d)]))/(b^2
*(a + b*x)^(1/6))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{1 \left ( dx+c \right ) ^{{\frac{7}{6}}} \left ( bx+a \right ) ^{-{\frac{7}{6}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(7/6)/(b*x+a)^(7/6),x)

[Out]

int((d*x+c)^(7/6)/(b*x+a)^(7/6),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{7}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(7/6)/(b*x + a)^(7/6),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(7/6)/(b*x + a)^(7/6), x)

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Fricas [A]  time = 0.271908, size = 3270, normalized size = 8.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(7/6)/(b*x + a)^(7/6),x, algorithm="fricas")

[Out]

1/12*(28*sqrt(3)*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*
d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(
1/6)*arctan(-sqrt(3)*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*
c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^1
3)^(1/6)/(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - 2*(b*x + a)*sqrt(((b^3
*c - a*b^2*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15
*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6
*d^7)/b^13)^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1
/3) + (b^5*x + a*b^4)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^
3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/3))/(b*x
+ a)) + (b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*
a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))) +
28*sqrt(3)*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 -
20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*a
rctan(-sqrt(3)*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^
3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/
6)/(2*(b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) - 2*(b*x + a)*sqrt(-((b^3*c -
a*b^2*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*
b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)
/b^13)^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) -
(b^5*x + a*b^4)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3
*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/3))/(b*x + a))
- (b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b
^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))) + 7*(b^
3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3
*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(49*((b^3*c
- a*b^2*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^
2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^
7)/b^13)^(1/6) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3)
+ (b^5*x + a*b^4)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b
^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/3))/(b*x + a
)) - 7*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a
^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-
49*((b^3*c - a*b^2*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6)*((b^6*c^6*d - 6*a*b^5*c^5*
d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d
^6 + a^6*d^7)/b^13)^(1/6) - (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(b*x + a)^(2/3)*(d*x
+ c)^(1/3) - (b^5*x + a*b^4)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3
- 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/3
))/(b*x + a)) + 14*(b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^
4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)
^(1/6)*log(-7*((b*c - a*d)*(b*x + a)^(5/6)*(d*x + c)^(1/6) + (b^3*x + a*b^2)*((b
^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^
2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6))/(b*x + a)) - 14*(b^3*x + a*b^2
)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a
^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d^7)/b^13)^(1/6)*log(-7*((b*c - a*d)*(b*x +
a)^(5/6)*(d*x + c)^(1/6) - (b^3*x + a*b^2)*((b^6*c^6*d - 6*a*b^5*c^5*d^2 + 15*a
^2*b^4*c^4*d^3 - 20*a^3*b^3*c^3*d^4 + 15*a^4*b^2*c^2*d^5 - 6*a^5*b*c*d^6 + a^6*d
^7)/b^13)^(1/6))/(b*x + a)) + 12*(b*d*x - 6*b*c + 7*a*d)*(b*x + a)^(5/6)*(d*x +
c)^(1/6))/(b^3*x + a*b^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((d*x+c)**(7/6)/(b*x+a)**(7/6),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{\frac{7}{6}}}{{\left (b x + a\right )}^{\frac{7}{6}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(7/6)/(b*x + a)^(7/6),x, algorithm="giac")

[Out]

integrate((d*x + c)^(7/6)/(b*x + a)^(7/6), x)