3.102 \(\int \frac{1}{(a+b x^3)^{4/3} (c+d x^3)^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac{d (3 b c-a d) \log \left (c+d x^3\right )}{9 c^{5/3} (b c-a d)^{7/3}}+\frac{d (3 b c-a d) \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} (b c-a d)^{7/3}}-\frac{2 d (3 b c-a d) \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^{7/3}}+\frac{b x (a d+3 b c)}{3 a c \sqrt [3]{a+b x^3} (b c-a d)^2}-\frac{d x}{3 c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)} \]

[Out]

(b*(3*b*c + a*d)*x)/(3*a*c*(b*c - a*d)^2*(a + b*x^3)^(1/3)) - (d*x)/(3*c*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*
x^3)) - (2*d*(3*b*c - a*d)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[
3]*c^(5/3)*(b*c - a*d)^(7/3)) - (d*(3*b*c - a*d)*Log[c + d*x^3])/(9*c^(5/3)*(b*c - a*d)^(7/3)) + (d*(3*b*c - a
*d)*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)])/(3*c^(5/3)*(b*c - a*d)^(7/3))

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Rubi [C]  time = 1.93369, antiderivative size = 625, normalized size of antiderivative = 2.39, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {430, 429} \[ -\frac{c \left (a+b x^3\right )^{2/3} \left (-\frac{54 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^5 \left (a+b x^3\right )^3}-\frac{108 d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac{54 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac{2520 d^2 x^9 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}-\frac{945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac{5320 d x^6 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}-\frac{1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}-\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}+\frac{2240 x^3 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-\frac{525 x^3 (b c-a d)}{c \left (a+b x^3\right )}+\frac{6300 d^2 x^6}{c^2}+\frac{13720 d x^3}{c}+6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(c*(a + b*x^3)^(2/3)*(6860 + (13720*d*x^3)/c + (6300*d^2*x^6)/c^2 - (525*(b*c - a*d)*x^3)/(c*(a + b*x^3)) - (
1890*d*(b*c - a*d)*x^6)/(c^2*(a + b*x^3)) - (945*d^2*(b*c - a*d)*x^9)/(c^3*(a + b*x^3)) - 6860*Hypergeometric2
F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - (13720*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*
x^3)/(c*(a + b*x^3))])/c - (6300*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/c^
2 + (2240*(b*c - a*d)*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c*(a + b*x^3)) +
(5320*d*(b*c - a*d)*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^2*(a + b*x^3))
+ (2520*d^2*(b*c - a*d)*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*(a + b*x^3
)) - (54*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*
(a + b*x^3)^3) - (108*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a +
b*x^3))])/(c^4*(a + b*x^3)^3) - (54*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c -
a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(a + b*x^3)^3)))/(420*(b*c - a*d)^2*x^5*(c + d*x^3))

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{4/3} \left (c+d x^3\right )^2} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac{c \left (a+b x^3\right )^{2/3} \left (6860+\frac{13720 d x^3}{c}+\frac{6300 d^2 x^6}{c^2}-\frac{525 (b c-a d) x^3}{c \left (a+b x^3\right )}-\frac{1890 d (b c-a d) x^6}{c^2 \left (a+b x^3\right )}-\frac{945 d^2 (b c-a d) x^9}{c^3 \left (a+b x^3\right )}-6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c}-\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2}+\frac{2240 (b c-a d) x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c \left (a+b x^3\right )}+\frac{5320 d (b c-a d) x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac{2520 d^2 (b c-a d) x^9 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )}-\frac{54 (b c-a d)^3 x^9 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )^3}-\frac{108 d (b c-a d)^3 x^{12} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^4 \left (a+b x^3\right )^3}-\frac{54 d^2 (b c-a d)^3 x^{15} \, _3F_2\left (2,2,\frac{7}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{c^5 \left (a+b x^3\right )^3}\right )}{420 (b c-a d)^2 x^5 \left (c+d x^3\right )}\\ \end{align*}

Mathematica [C]  time = 1.45104, size = 625, normalized size = 2.39 \[ \frac{c \left (a+b x^3\right )^{2/3} \left (\frac{54 d^2 x^{15} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^5 \left (a+b x^3\right )^3}+\frac{108 d x^{12} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^4 \left (a+b x^3\right )^3}+\frac{54 x^9 (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )}{c^3 \left (a+b x^3\right )^3}+\frac{2520 d^2 x^9 (a d-b c) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^3 \left (a+b x^3\right )}+\frac{6300 d^2 x^6 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2}+\frac{945 d^2 x^9 (b c-a d)}{c^3 \left (a+b x^3\right )}+\frac{5320 d x^6 (a d-b c) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c^2 \left (a+b x^3\right )}+\frac{1890 d x^6 (b c-a d)}{c^2 \left (a+b x^3\right )}+\frac{13720 d x^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c}-\frac{2240 x^3 (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{c \left (a+b x^3\right )}+6860 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+\frac{525 x^3 (b c-a d)}{c \left (a+b x^3\right )}-\frac{6300 d^2 x^6}{c^2}-\frac{13720 d x^3}{c}-6860\right )}{420 x^5 \left (c+d x^3\right ) (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*(a + b*x^3)^(2/3)*(-6860 - (13720*d*x^3)/c - (6300*d^2*x^6)/c^2 + (525*(b*c - a*d)*x^3)/(c*(a + b*x^3)) + (
1890*d*(b*c - a*d)*x^6)/(c^2*(a + b*x^3)) + (945*d^2*(b*c - a*d)*x^9)/(c^3*(a + b*x^3)) + 6860*Hypergeometric2
F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + (13720*d*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*
x^3)/(c*(a + b*x^3))])/c + (6300*d^2*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/c^
2 - (2240*(b*c - a*d)*x^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c*(a + b*x^3)) +
(5320*d*(-(b*c) + a*d)*x^6*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^2*(a + b*x^3
)) + (2520*d^2*(-(b*c) + a*d)*x^9*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^3*(a +
b*x^3)) + (54*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])
/(c^3*(a + b*x^3)^3) + (108*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(
c*(a + b*x^3))])/(c^4*(a + b*x^3)^3) + (54*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 7/3}, {1, 13/3}, ((
b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(a + b*x^3)^3)))/(420*(b*c - a*d)^2*x^5*(c + d*x^3))

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Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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