3.3020 \(\int \frac{(a+b x)^{4/3} (e+f x)^2}{(c+d x)^{4/3}} \, dx\)

Optimal. Leaf size=562 \[ -\frac{4 \sqrt [3]{a+b x} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{27 b d^4}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{9 b d^3 (b c-a d)}-\frac{2 (b c-a d) \log (a+b x) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{81 b^{5/3} d^{13/3}}-\frac{2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{27 b^{5/3} d^{13/3}}-\frac{4 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{5/3} d^{13/3}}+\frac{3 (a+b x)^{7/3} (d e-c f)^2}{d^2 \sqrt [3]{c+d x} (b c-a d)}+\frac{f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2} \]

[Out]

(3*(d*e - c*f)^2*(a + b*x)^(7/3))/(d^2*(b*c - a*d)*(c + d*x)^(1/3)) - (4*(a^2*d^
2*f^2 - a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(27*b*d^4) + ((a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c
*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*(a + b*x)^(4/3)*(c + d*x)^(2/3
))/(9*b*d^3*(b*c - a*d)) + (f^2*(a + b*x)^(7/3)*(c + d*x)^(2/3))/(3*b*d^2) - (4*
(b*c - a*d)*(a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*
f + 35*c^2*f^2))*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)
*(a + b*x)^(1/3))])/(27*Sqrt[3]*b^(5/3)*d^(13/3)) - (2*(b*c - a*d)*(a^2*d^2*f^2
- a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*Log[a +
b*x])/(81*b^(5/3)*d^(13/3)) - (2*(b*c - a*d)*(a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c
*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1
/3))/(d^(1/3)*(a + b*x)^(1/3))])/(27*b^(5/3)*d^(13/3))

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Rubi [A]  time = 1.51129, antiderivative size = 562, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{4 \sqrt [3]{a+b x} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{27 b d^4}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{9 b d^3 (b c-a d)}-\frac{2 (b c-a d) \log (a+b x) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right )}{81 b^{5/3} d^{13/3}}-\frac{2 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{27 b^{5/3} d^{13/3}}-\frac{4 (b c-a d) \left (a^2 d^2 f^2-a b d f (9 d e-7 c f)+b^2 \left (-\left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{5/3} d^{13/3}}+\frac{3 (a+b x)^{7/3} (d e-c f)^2}{d^2 \sqrt [3]{c+d x} (b c-a d)}+\frac{f^2 (a+b x)^{7/3} (c+d x)^{2/3}}{3 b d^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*(d*e - c*f)^2*(a + b*x)^(7/3))/(d^2*(b*c - a*d)*(c + d*x)^(1/3)) - (4*(a^2*d^
2*f^2 - a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(27*b*d^4) + ((a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c
*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*(a + b*x)^(4/3)*(c + d*x)^(2/3
))/(9*b*d^3*(b*c - a*d)) + (f^2*(a + b*x)^(7/3)*(c + d*x)^(2/3))/(3*b*d^2) - (4*
(b*c - a*d)*(a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*
f + 35*c^2*f^2))*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)
*(a + b*x)^(1/3))])/(27*Sqrt[3]*b^(5/3)*d^(13/3)) - (2*(b*c - a*d)*(a^2*d^2*f^2
- a*b*d*f*(9*d*e - 7*c*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*Log[a +
b*x])/(81*b^(5/3)*d^(13/3)) - (2*(b*c - a*d)*(a^2*d^2*f^2 - a*b*d*f*(9*d*e - 7*c
*f) - b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*Log[-1 + (b^(1/3)*(c + d*x)^(1
/3))/(d^(1/3)*(a + b*x)^(1/3))])/(27*b^(5/3)*d^(13/3))

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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((b*x+a)**(4/3)*(f*x+e)**2/(d*x+c)**(4/3),x)

[Out]

Timed out

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Mathematica [C]  time = 0.916456, size = 282, normalized size = 0.5 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{2 \left (-a^2 d^2 f^2+a b d f (9 d e-7 c f)+b^2 \left (35 c^2 f^2-63 c d e f+27 d^2 e^2\right )\right ) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{b (c+d x)}{b c-a d}\right )}{\sqrt [3]{\frac{d (a+b x)}{a d-b c}}}+\frac{2 a^2 d^2 f^2 (c+d x)+a b d \left (-133 c^2 f^2+c d f (225 e-37 f x)+d^2 \left (-81 e^2+63 e f x+15 f^2 x^2\right )\right )+b^2 \left (140 c^3 f^2+7 c^2 d f (5 f x-36 e)+3 c d^2 \left (36 e^2-21 e f x-5 f^2 x^2\right )+9 d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )\right )}{c+d x}\right )}{27 b d^4} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3)*((2*a^2*d^2*f^2*(c + d*x) + b^2*(140*c^3*f^2 +
7*c^2*d*f*(-36*e + 5*f*x) + 3*c*d^2*(36*e^2 - 21*e*f*x - 5*f^2*x^2) + 9*d^3*x*(3
*e^2 + 3*e*f*x + f^2*x^2)) + a*b*d*(-133*c^2*f^2 + c*d*f*(225*e - 37*f*x) + d^2*
(-81*e^2 + 63*e*f*x + 15*f^2*x^2)))/(c + d*x) + (2*(-(a^2*d^2*f^2) + a*b*d*f*(9*
d*e - 7*c*f) + b^2*(27*d^2*e^2 - 63*c*d*e*f + 35*c^2*f^2))*Hypergeometric2F1[2/3
, 2/3, 5/3, (b*(c + d*x))/(b*c - a*d)])/((d*(a + b*x))/(-(b*c) + a*d))^(1/3)))/(
27*b*d^4)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{ \left ( fx+e \right ) ^{2} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.317482, size = 1403, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/243*sqrt(3)*(3*sqrt(3)*(9*b^2*d^3*f^2*x^3 + 27*(4*b^2*c*d^2 - 3*a*b*d^3)*e^2 -
9*(28*b^2*c^2*d - 25*a*b*c*d^2)*e*f + (140*b^2*c^3 - 133*a*b*c^2*d + 2*a^2*c*d^
2)*f^2 + 3*(9*b^2*d^3*e*f - 5*(b^2*c*d^2 - a*b*d^3)*f^2)*x^2 + (27*b^2*d^3*e^2 -
63*(b^2*c*d^2 - a*b*d^3)*e*f + (35*b^2*c^2*d - 37*a*b*c*d^2 + 2*a^2*d^3)*f^2)*x
)*(b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - 2*sqrt(3)*(27*(b^3*c^2*d^2 - a
*b^2*c*d^3)*e^2 - 9*(7*b^3*c^3*d - 8*a*b^2*c^2*d^2 + a^2*b*c*d^3)*e*f + (35*b^3*
c^4 - 42*a*b^2*c^3*d + 6*a^2*b*c^2*d^2 + a^3*c*d^3)*f^2 + (27*(b^3*c*d^3 - a*b^2
*d^4)*e^2 - 9*(7*b^3*c^2*d^2 - 8*a*b^2*c*d^3 + a^2*b*d^4)*e*f + (35*b^3*c^3*d -
42*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3 + a^3*d^4)*f^2)*x)*log((b^2*d*x + b^2*c + (b^2*
d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b + (b^2*d)^(2/3)*(b*x + a)^(2/3)*(d*x
+ c)^(1/3))/(d*x + c)) + 4*sqrt(3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^2 - 9*(7*b^
3*c^3*d - 8*a*b^2*c^2*d^2 + a^2*b*c*d^3)*e*f + (35*b^3*c^4 - 42*a*b^2*c^3*d + 6*
a^2*b*c^2*d^2 + a^3*c*d^3)*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2 - 9*(7*b^3*c^2*
d^2 - 8*a*b^2*c*d^3 + a^2*b*d^4)*e*f + (35*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 6*a^2*
b*c*d^3 + a^3*d^4)*f^2)*x)*log(-(b*d*x + b*c - (b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*
x + c)^(2/3))/(d*x + c)) - 12*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^2 - 9*(7*b^3*c^3
*d - 8*a*b^2*c^2*d^2 + a^2*b*c*d^3)*e*f + (35*b^3*c^4 - 42*a*b^2*c^3*d + 6*a^2*b
*c^2*d^2 + a^3*c*d^3)*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2 - 9*(7*b^3*c^2*d^2 -
8*a*b^2*c*d^3 + a^2*b*d^4)*e*f + (35*b^3*c^3*d - 42*a*b^2*c^2*d^2 + 6*a^2*b*c*d
^3 + a^3*d^4)*f^2)*x)*arctan(1/3*(2*sqrt(3)*(b^2*d)^(1/3)*(b*x + a)^(1/3)*(d*x +
c)^(2/3) + sqrt(3)*(b*d*x + b*c))/(b*d*x + b*c)))/((b*d^5*x + b*c*d^4)*(b^2*d)^
(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError