3.83 \(\int \frac{(c+d x^3)^2}{(a+b x^3)^{5/3}} \, dx\)

Optimal. Leaf size=146 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac{d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0951581, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 388, 246, 245} \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-2 a^2 d^2+2 a b c d+b^2 c^2\right ) \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}-\frac{d x \sqrt [3]{a+b x^3} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^3\right ) (b c-a d)}{2 a b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(5/3),x]

[Out]

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

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{5/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac{\int \frac{c (b c+a d)-2 d (b c-2 a d) x^3}{\left (a+b x^3\right )^{2/3}} \, dx}{2 a b}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac{(-2 a d (b c-2 a d)-2 b c (b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{2/3}} \, dx}{4 a b^2}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+-\frac{\left ((-2 a d (b c-2 a d)-2 b c (b c+a d)) \left (1+\frac{b x^3}{a}\right )^{2/3}\right ) \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{2/3}} \, dx}{4 a b^2 \left (a+b x^3\right )^{2/3}}\\ &=-\frac{d (b c-2 a d) x \sqrt [3]{a+b x^3}}{2 a b^2}+\frac{(b c-a d) x \left (c+d x^3\right )}{2 a b \left (a+b x^3\right )^{2/3}}+\frac{\left (b^2 c^2+2 a b c d-2 a^2 d^2\right ) x \left (1+\frac{b x^3}{a}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{2 a b^2 \left (a+b x^3\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 2.52533, size = 171, normalized size = 1.17 \[ \frac{x \text{Gamma}\left (\frac{2}{3}\right ) \left (\frac{b x^3}{a}+1\right )^{2/3} \left (-3 b x^3 \left (c+d x^3\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},2,\frac{8}{3}\right \},\left \{1,\frac{13}{3}\right \},-\frac{b x^3}{a}\right )-b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \, _2F_1\left (\frac{4}{3},\frac{8}{3};\frac{13}{3};-\frac{b x^3}{a}\right )+4 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \, _2F_1\left (\frac{1}{3},\frac{5}{3};\frac{10}{3};-\frac{b x^3}{a}\right )\right )}{84 a^2 \text{Gamma}\left (\frac{5}{3}\right ) \left (a+b x^3\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(5/3),x]

[Out]

(x*(1 + (b*x^3)/a)^(2/3)*Gamma[2/3]*(4*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Hypergeometric2F1[1/3, 5/3, 10/3, -(
(b*x^3)/a)] - b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2*x^6)*Hypergeometric2F1[4/3, 8/3, 13/3, -((b*x^3)/a)] - 3*b*x^
3*(c + d*x^3)^2*HypergeometricPFQ[{4/3, 2, 8/3}, {1, 13/3}, -((b*x^3)/a)]))/(84*a^2*(a + b*x^3)^(2/3)*Gamma[5/
3])

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Maple [F]  time = 0.427, size = 0, normalized size = 0. \begin{align*} \int{ \left ( d{x}^{3}+c \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{-{\frac{5}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(5/3),x)

[Out]

int((d*x^3+c)^2/(b*x^3+a)^(5/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(5/3),x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(5/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d^{2} x^{6} + 2 \, c d x^{3} + c^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(5/3),x, algorithm="fricas")

[Out]

integral((d^2*x^6 + 2*c*d*x^3 + c^2)*(b*x^3 + a)^(1/3)/(b^2*x^6 + 2*a*b*x^3 + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac{5}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**2/(b*x**3+a)**(5/3),x)

[Out]

Integral((c + d*x**3)**2/(a + b*x**3)**(5/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(5/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(5/3), x)