3.3012 \(\int \frac{(a+b x)^3}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx\)

Optimal. Leaf size=1388 \[ \text{result too large to display} \]

[Out]

(3*(a + b*x)^2*(c + d*x)^(2/3))/(14*d^2*(b*c + a*d + 2*b*d*x)^(1/3)) + (9*(b*c -
a*d)*(c + d*x)^(2/3)*(b*c - 7*a*d - 6*b*d*x))/(112*d^4*(b*c + a*d + 2*b*d*x)^(1
/3)) + (81*(b*c - a*d)^2*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[d^2*(3*b*c
+ a*d + 4*b*d*x)^2]*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(112*b^(2/3)*d^6*(c
+ d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*((1 + Sqrt[3])*
(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))) - (81*3^
(1/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3
)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x
)*(a*d + b*(c + 2*d*x)))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^
(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c
+ 2*d*x)))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d
+ b*(c + 2*d*x)))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) +
2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2
/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(224
*b^(2/3)*d^4*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)
*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3)
+ 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)
^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]) + (27*3^(3/4)*(b
*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d) +
4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(
1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d + b*
(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + S
qrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2
]*EllipticF[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d
+ b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)
*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(56*Sqrt[2]*b^(2/3)*d^4*(c + d
*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c +
a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d
*x)*(a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*
((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2])

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Rubi [A]  time = 4.96443, antiderivative size = 1388, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212 \[ -\frac{81 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt{3}\right ) (b c-a d)^{8/3}}{224 b^{2/3} d^4 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac{27\ 3^{3/4} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))} (b c-a d)^{2/3}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt{3}\right ) (b c-a d)^{8/3}}{56 \sqrt{2} b^{2/3} d^4 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac{81 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt{d^2 (3 b c+a d+4 b d x)^2} \sqrt{\left (4 b x d^2+(3 b c+a d) d\right )^2} (b c-a d)^2}{112 b^{2/3} d^6 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}+\frac{9 (c+d x)^{2/3} (b c-7 a d-6 b d x) (b c-a d)}{112 d^4 \sqrt [3]{b c+a d+2 b d x}}+\frac{3 (a+b x)^2 (c+d x)^{2/3}}{14 d^2 \sqrt [3]{b c+a d+2 b d x}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(a + b*x)^2*(c + d*x)^(2/3))/(14*d^2*(b*c + a*d + 2*b*d*x)^(1/3)) + (9*(b*c -
a*d)*(c + d*x)^(2/3)*(b*c - 7*a*d - 6*b*d*x))/(112*d^4*(b*c + a*d + 2*b*d*x)^(1
/3)) + (81*(b*c - a*d)^2*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[d^2*(3*b*c
+ a*d + 4*b*d*x)^2]*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2])/(112*b^(2/3)*d^6*(c
+ d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*((1 + Sqrt[3])*
(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))) - (81*3^
(1/4)*Sqrt[2 - Sqrt[3]]*(b*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3
)*Sqrt[(d*(3*b*c + a*d) + 4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x
)*(a*d + b*(c + 2*d*x)))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^
(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c
+ 2*d*x)))^(2/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d
+ b*(c + 2*d*x)))^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) +
2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2
/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(224
*b^(2/3)*d^4*(c + d*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)
*Sqrt[d^2*(3*b*c + a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3)
+ 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)
^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2]) + (27*3^(3/4)*(b
*c - a*d)^(8/3)*((c + d*x)*(b*c + a*d + 2*b*d*x))^(1/3)*Sqrt[(d*(3*b*c + a*d) +
4*b*d^2*x)^2]*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(
1/3))*Sqrt[((b*c - a*d)^(4/3) - 2*b^(1/3)*(b*c - a*d)^(2/3)*((c + d*x)*(a*d + b*
(c + 2*d*x)))^(1/3) + 4*b^(2/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(2/3))/((1 + S
qrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2
]*EllipticF[ArcSin[((1 - Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)*(a*d
+ b*(c + 2*d*x)))^(1/3))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d*x)
*(a*d + b*(c + 2*d*x)))^(1/3))], -7 - 4*Sqrt[3]])/(56*Sqrt[2]*b^(2/3)*d^4*(c + d
*x)^(1/3)*(b*c + a*d + 2*b*d*x)^(1/3)*(3*b*c + a*d + 4*b*d*x)*Sqrt[d^2*(3*b*c +
a*d + 4*b*d*x)^2]*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2*b^(1/3)*((c + d
*x)*(a*d + b*(c + 2*d*x)))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2*b^(1/3)*
((c + d*x)*(a*d + b*(c + 2*d*x)))^(1/3))^2])

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

[Out]

Timed out

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Mathematica [C]  time = 0.602512, size = 157, normalized size = 0.11 \[ \frac{(a d+b (c+2 d x))^{2/3} \left (\frac{81 \sqrt [3]{2} (b c-a d)^2 \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{a d+b (c+2 d x)}{a d-b c}\right )}{b d^4}+\frac{6 (c+d x) \left (\frac{14 (b c-a d)^2}{a d+b (c+2 d x)}+15 a d-11 b c+4 b d x\right )}{d^4}\right )}{224 \sqrt [3]{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

((a*d + b*(c + 2*d*x))^(2/3)*((6*(c + d*x)*(-11*b*c + 15*a*d + 4*b*d*x + (14*(b*
c - a*d)^2)/(a*d + b*(c + 2*d*x))))/d^4 + (81*2^(1/3)*(b*c - a*d)^2*((b*(c + d*x
))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, (a*d + b*(c + 2*d*x))/(-(
b*c) + a*d)])/(b*d^4)))/(224*(c + d*x)^(1/3))

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{ \left ( bx+a \right ) ^{3}{\frac{1}{\sqrt [3]{dx+c}}} \left ( 2\,bdx+ad+bc \right ) ^{-{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError