3.1609 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{2/3}} \, dx\)

Optimal. Leaf size=614 \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}}{2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}}\right ),-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]

[Out]

(-6*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(5*d^2) + (3*(a + b*x)^(4/3)*(c + d*x)^(1/3))/(5*d) + (2*2^(1
/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(2/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c -
a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)*b^(1/3)*d^
(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/(
(1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[ArcSin[((1
- Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d
)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(5*b^(1/3)*d^(7/3)*(a + b*x)
^(2/3)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1
/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d
*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])

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Rubi [A]  time = 0.817457, antiderivative size = 614, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 62, 623, 218} \[ \frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(a d+b c+2 b d x)^2} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right ) \sqrt{\frac{2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{4/3}}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (a d+b c+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+(b c-a d)^{2/3}\right )}{\left (2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}+\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}-\frac{6 \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c-a d)}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(b*c - a*d)*(a + b*x)^(1/3)*(c + d*x)^(1/3))/(5*d^2) + (3*(a + b*x)^(4/3)*(c + d*x)^(1/3))/(5*d) + (2*2^(1
/3)*3^(3/4)*Sqrt[2 + Sqrt[3]]*(b*c - a*d)^2*((a + b*x)*(c + d*x))^(2/3)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*((b*c -
a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))*Sqrt[((b*c - a*d)^(4/3) - 2^(2/3)*b^(1/3)*d^
(1/3)*(b*c - a*d)^(2/3)*((a + b*x)*(c + d*x))^(1/3) + 2*2^(1/3)*b^(2/3)*d^(2/3)*((a + b*x)*(c + d*x))^(2/3))/(
(1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))^2]*EllipticF[ArcSin[((1
- Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))/((1 + Sqrt[3])*(b*c - a*d
)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d*x))^(1/3))], -7 - 4*Sqrt[3]])/(5*b^(1/3)*d^(7/3)*(a + b*x)
^(2/3)*(c + d*x)^(2/3)*(b*c + a*d + 2*b*d*x)*Sqrt[((b*c - a*d)^(2/3)*((b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1
/3)*((a + b*x)*(c + d*x))^(1/3)))/((1 + Sqrt[3])*(b*c - a*d)^(2/3) + 2^(2/3)*b^(1/3)*d^(1/3)*((a + b*x)*(c + d
*x))^(1/3))^2]*Sqrt[(a*d + b*(c + 2*d*x))^2])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 62

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/((a + b*x)
*(c + d*x))^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] &&
LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{(a+b x)^{4/3}}{(c+d x)^{2/3}} \, dx &=\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d}-\frac{(4 (b c-a d)) \int \frac{\sqrt [3]{a+b x}}{(c+d x)^{2/3}} \, dx}{5 d}\\ &=-\frac{6 (b c-a d) \sqrt [3]{a+b x} \sqrt [3]{c+d x}}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d}+\frac{\left (2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^{2/3} (c+d x)^{2/3}} \, dx}{5 d^2}\\ &=-\frac{6 (b c-a d) \sqrt [3]{a+b x} \sqrt [3]{c+d x}}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d}+\frac{\left (2 (b c-a d)^2 ((a+b x) (c+d x))^{2/3}\right ) \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^{2/3}} \, dx}{5 d^2 (a+b x)^{2/3} (c+d x)^{2/3}}\\ &=-\frac{6 (b c-a d) \sqrt [3]{a+b x} \sqrt [3]{c+d x}}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d}+\frac{\left (6 (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^3}} \, dx,x,\sqrt [3]{(a+b x) (c+d x)}\right )}{5 d^2 (a+b x)^{2/3} (c+d x)^{2/3} (b c+a d+2 b d x)}\\ &=-\frac{6 (b c-a d) \sqrt [3]{a+b x} \sqrt [3]{c+d x}}{5 d^2}+\frac{3 (a+b x)^{4/3} \sqrt [3]{c+d x}}{5 d}+\frac{2 \sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} (b c-a d)^2 ((a+b x) (c+d x))^{2/3} \sqrt{(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt{\frac{(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt{3}\right )}{5 \sqrt [3]{b} d^{7/3} (a+b x)^{2/3} (c+d x)^{2/3} (b c+a d+2 b d x) \sqrt{\frac{(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt{3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt{(a d+b (c+2 d x))^2}}\\ \end{align*}

Mathematica [C]  time = 0.028695, size = 73, normalized size = 0.12 \[ \frac{3 (a+b x)^{7/3} \left (\frac{b (c+d x)}{b c-a d}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{7}{3};\frac{10}{3};\frac{d (a+b x)}{a d-b c}\right )}{7 b (c+d x)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + b*x)^(7/3)*((b*(c + d*x))/(b*c - a*d))^(2/3)*Hypergeometric2F1[2/3, 7/3, 10/3, (d*(a + b*x))/(-(b*c) +
a*d)])/(7*b*(c + d*x)^(2/3))

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Maple [F]  time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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