3.91 \(\int \frac{1}{(a+b x^3)^{7/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=226 \[ \frac{b x (3 b c-7 a d)}{4 a^2 \sqrt [3]{a+b x^3} (b c-a d)^2}+\frac{d^2 \log \left (c+d x^3\right )}{6 c^{2/3} (b c-a d)^{7/3}}-\frac{d^2 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} (b c-a d)^{7/3}}+\frac{d^2 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} c^{2/3} (b c-a d)^{7/3}}+\frac{b x}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)} \]

[Out]

(b*x)/(4*a*(b*c - a*d)*(a + b*x^3)^(4/3)) + (b*(3*b*c - 7*a*d)*x)/(4*a^2*(b*c - a*d)^2*(a + b*x^3)^(1/3)) + (d
^2*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]])/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(7/
3)) + (d^2*Log[c + d*x^3])/(6*c^(2/3)*(b*c - a*d)^(7/3)) - (d^2*Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3
)^(1/3)])/(2*c^(2/3)*(b*c - a*d)^(7/3))

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Rubi [C]  time = 2.58034, antiderivative size = 621, normalized size of antiderivative = 2.75, 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} \[ -\frac{-9 c^2 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-9 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-18 c d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-180 c^3 d^2 x^6 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+45 c^2 d^2 x^9 \left (a+b x^3\right )^2 (b c-a d)+180 c^3 d^2 x^6 \left (a+b x^3\right )^3-33 c^2 x^9 (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-420 c^4 d x^3 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-280 c^5 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+105 c^3 d x^6 \left (a+b x^3\right )^2 (b c-a d)+420 c^4 d x^3 \left (a+b x^3\right )^3+70 c^4 x^3 \left (a+b x^3\right )^2 (b c-a d)+280 c^5 \left (a+b x^3\right )^3-27 d^2 x^{15} (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-60 c d x^{12} (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{40 c^4 x^5 \left (a+b x^3\right )^{10/3} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/((a + b*x^3)^(7/3)*(c + d*x^3)),x]

[Out]

-(70*c^4*(b*c - a*d)*x^3*(a + b*x^3)^2 + 105*c^3*d*(b*c - a*d)*x^6*(a + b*x^3)^2 + 45*c^2*d^2*(b*c - a*d)*x^9*
(a + b*x^3)^2 + 280*c^5*(a + b*x^3)^3 + 420*c^4*d*x^3*(a + b*x^3)^3 + 180*c^3*d^2*x^6*(a + b*x^3)^3 - 280*c^5*
(a + b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 420*c^4*d*x^3*(a + b*x^3)^3*
Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 180*c^3*d^2*x^6*(a + b*x^3)^3*Hypergeometr
ic2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 33*c^2*(b*c - a*d)^3*x^9*Hypergeometric2F1[2, 10/3, 13
/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] - 60*c*d*(b*c - a*d)^3*x^12*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*
d)*x^3)/(c*(a + b*x^3))] - 27*d^2*(b*c - a*d)^3*x^15*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] - 9*c^2*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^
3))] - 18*c*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
- 9*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(40
*c^4*(b*c - a*d)^2*x^5*(a + b*x^3)^(10/3))

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 )^{7/3} \left (c+d x^3\right )} \, dx &=\frac{\sqrt [3]{1+\frac{b x^3}{a}} \int \frac{1}{\left (1+\frac{b x^3}{a}\right )^{7/3} \left (c+d x^3\right )} \, dx}{a^2 \sqrt [3]{a+b x^3}}\\ &=-\frac{70 c^4 (b c-a d) x^3 \left (a+b x^3\right )^2+105 c^3 d (b c-a d) x^6 \left (a+b x^3\right )^2+45 c^2 d^2 (b c-a d) x^9 \left (a+b x^3\right )^2+280 c^5 \left (a+b x^3\right )^3+420 c^4 d x^3 \left (a+b x^3\right )^3+180 c^3 d^2 x^6 \left (a+b x^3\right )^3-280 c^5 \left (a+b x^3\right )^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 )-420 c^4 d x^3 \left (a+b x^3\right )^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 )-180 c^3 d^2 x^6 \left (a+b x^3\right )^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 )-33 c^2 (b c-a d)^3 x^9 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-60 c d (b c-a d)^3 x^{12} \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-27 d^2 (b c-a d)^3 x^{15} \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-9 c^2 (b c-a d)^3 x^9 \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-18 c d (b c-a d)^3 x^{12} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )-9 d^2 (b c-a d)^3 x^{15} \, _3F_2\left (2,2,\frac{10}{3};1,\frac{13}{3};\frac{(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{40 c^4 (b c-a d)^2 x^5 \left (a+b x^3\right )^{10/3}}\\ \end{align*}

Mathematica [C]  time = 1.92663, size = 621, normalized size = 2.75 \[ \frac{9 c^2 x^9 (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{10}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )+9 d^2 x^{15} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{10}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )+18 c d x^{12} (b c-a d)^3 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{10}{3}\right \},\left \{1,\frac{13}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )+180 c^3 d^2 x^6 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-45 c^2 d^2 x^9 \left (a+b x^3\right )^2 (b c-a d)-180 c^3 d^2 x^6 \left (a+b x^3\right )^3+33 c^2 x^9 (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+420 c^4 d x^3 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+280 c^5 \left (a+b x^3\right )^3 \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-105 c^3 d x^6 \left (a+b x^3\right )^2 (b c-a d)-420 c^4 d x^3 \left (a+b x^3\right )^3-70 c^4 x^3 \left (a+b x^3\right )^2 (b c-a d)-280 c^5 \left (a+b x^3\right )^3+27 d^2 x^{15} (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+60 c d x^{12} (b c-a d)^3 \, _2F_1\left (2,\frac{10}{3};\frac{13}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{40 c^4 x^5 \left (a+b x^3\right )^{10/3} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(7/3)*(c + d*x^3)),x]

[Out]

(-70*c^4*(b*c - a*d)*x^3*(a + b*x^3)^2 - 105*c^3*d*(b*c - a*d)*x^6*(a + b*x^3)^2 - 45*c^2*d^2*(b*c - a*d)*x^9*
(a + b*x^3)^2 - 280*c^5*(a + b*x^3)^3 - 420*c^4*d*x^3*(a + b*x^3)^3 - 180*c^3*d^2*x^6*(a + b*x^3)^3 + 280*c^5*
(a + b*x^3)^3*Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 420*c^4*d*x^3*(a + b*x^3)^3*
Hypergeometric2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 180*c^3*d^2*x^6*(a + b*x^3)^3*Hypergeometr
ic2F1[1/3, 1, 4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 33*c^2*(b*c - a*d)^3*x^9*Hypergeometric2F1[2, 10/3, 13
/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))] + 60*c*d*(b*c - a*d)^3*x^12*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*
d)*x^3)/(c*(a + b*x^3))] + 27*d^2*(b*c - a*d)^3*x^15*Hypergeometric2F1[2, 10/3, 13/3, ((b*c - a*d)*x^3)/(c*(a
+ b*x^3))] + 9*c^2*(b*c - a*d)^3*x^9*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^
3))] + 18*c*d*(b*c - a*d)^3*x^12*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]
+ 9*d^2*(b*c - a*d)^3*x^15*HypergeometricPFQ[{2, 2, 10/3}, {1, 13/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(40
*c^4*(b*c - a*d)^2*x^5*(a + b*x^3)^(10/3))

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Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{3}+c} \left ( b{x}^{3}+a \right ) ^{-{\frac{7}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(7/3)/(d*x^3+c),x)

[Out]

int(1/(b*x^3+a)^(7/3)/(d*x^3+c),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{7}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(7/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)), 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)^(7/3)/(d*x^3+c),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{3}\right )^{\frac{7}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a)**(7/3)/(d*x**3+c),x)

[Out]

Integral(1/((a + b*x**3)**(7/3)*(c + d*x**3)), x)

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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{7}{3}}{\left (d x^{3} + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(7/3)/(d*x^3+c),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)), x)