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]

1/4*b*x/a/(-a*d+b*c)/(b*x^3+a)^(4/3)+1/4*b*(-7*a*d+3*b*c)*x/a^2/(-a*d+b*c)^2/(b*x^3+a)^(1/3)+1/6*d^2*ln(d*x^3+
c)/c^(2/3)/(-a*d+b*c)^(7/3)-1/2*d^2*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(2/3)/(-a*d+b*c)^(7/3)+1/
3*d^2*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(2/3)/(-a*d+b*c)^(7/3)*3^(1/2)

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Rubi [C]  time = 2.58, 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 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])

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])

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*}

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Mathematica [C]  time = 2.67, size = 621, normalized size = 2.75 \[ \frac {9 c^2 x^9 (b c-a d)^3 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+9 d^2 x^{15} (b c-a d)^3 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+18 c d x^{12} (b c-a d)^3 \, _3F_2\left (2,2,\frac {10}{3};1,\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+280 c^5 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-280 c^5 \left (a+b x^3\right )^3+420 c^4 d x^3 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-420 c^4 d x^3 \left (a+b x^3\right )^3-70 c^4 x^3 \left (a+b x^3\right )^2 (b c-a d)+180 c^3 d^2 x^6 \left (a+b x^3\right )^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-180 c^3 d^2 x^6 \left (a+b x^3\right )^3-105 c^3 d x^6 \left (a+b x^3\right )^2 (b c-a d)-45 c^2 d^2 x^9 \left (a+b x^3\right )^2 (b c-a d)+33 c^2 x^9 (b c-a d)^3 \, _2F_1\left (2,\frac {10}{3};\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+27 d^2 x^{15} (b c-a d)^3 \, _2F_1\left (2,\frac {10}{3};\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+60 c d x^{12} (b c-a d)^3 \, _2F_1\left (2,\frac {10}{3};\frac {13}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{40 c^4 x^5 \left (a+b x^3\right )^{10/3} (b c-a d)^2} \]

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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]

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)

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maple [F]  time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {7}{3}} \left (d \,x^{3}+c \right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,\left (d\,x^3+c\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {7}{3}} \left (c + d x^{3}\right )}\, dx \]

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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