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

Optimal. Leaf size=377 \[ \frac {d x \left (a+b x^3\right )^{2/3} \left (-5 a^2 d^2+15 a b c d+18 b^2 c^2\right )}{18 a c^2 \left (c+d x^3\right ) (b c-a d)^3}-\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{10/3}}+\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{10/3}}-\frac {d \left (5 a^2 d^2-18 a b c d+27 b^2 c^2\right ) \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{10/3}}+\frac {b x (a d+6 b c)}{6 a c \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)^2}-\frac {d x}{6 c \sqrt [3]{a+b x^3} \left (c+d x^3\right )^2 (b c-a d)} \]

[Out]

-1/6*d*x/c/(-a*d+b*c)/(b*x^3+a)^(1/3)/(d*x^3+c)^2+1/6*b*(a*d+6*b*c)*x/a/c/(-a*d+b*c)^2/(b*x^3+a)^(1/3)/(d*x^3+
c)+1/18*d*(-5*a^2*d^2+15*a*b*c*d+18*b^2*c^2)*x*(b*x^3+a)^(2/3)/a/c^2/(-a*d+b*c)^3/(d*x^3+c)-1/54*d*(5*a^2*d^2-
18*a*b*c*d+27*b^2*c^2)*ln(d*x^3+c)/c^(8/3)/(-a*d+b*c)^(10/3)+1/18*d*(5*a^2*d^2-18*a*b*c*d+27*b^2*c^2)*ln((-a*d
+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(8/3)/(-a*d+b*c)^(10/3)-1/27*d*(5*a^2*d^2-18*a*b*c*d+27*b^2*c^2)*arct
an(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(8/3)/(-a*d+b*c)^(10/3)*3^(1/2)

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Rubi [C]  time = 2.74, antiderivative size = 428, normalized size of antiderivative = 1.14, 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} \[ -\frac {65 c^2 \left (a+b x^3\right )^2 \left (-28 \left (c+d x^3\right )^2 \left (a^2 \left (843 c^2 d x^3+500 c^3+375 c d^2 x^6+27 d^3 x^9\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+33657 a^2 c^2 d^3 x^9+60807 a^2 c^3 d^2 x^6+48104 a^2 c^4 d x^3+14000 a^2 c^5+7155 a^2 c d^4 x^{12}+243 a^2 d^5 x^{15}+38652 a b c^2 d^3 x^{12}+81534 a b c^3 d^2 x^9+70802 a b c^4 d x^6+21896 a b c^5 x^3+5940 a b c d^4 x^{15}+7425 b^2 c^2 d^3 x^{15}+23409 b^2 c^3 d^2 x^{12}+24417 b^2 c^4 d x^9+8391 b^2 c^5 x^6\right )-486 x^{12} \left (c+d x^3\right )^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )}{16380 c^5 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 (b c-a d)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(65*c^2*(a + b*x^3)^2*(14000*a^2*c^5 + 21896*a*b*c^5*x^3 + 48104*a^2*c^4*d*x^3 + 8391*b^2*c^5*x^6 + 70802*a*b
*c^4*d*x^6 + 60807*a^2*c^3*d^2*x^6 + 24417*b^2*c^4*d*x^9 + 81534*a*b*c^3*d^2*x^9 + 33657*a^2*c^2*d^3*x^9 + 234
09*b^2*c^3*d^2*x^12 + 38652*a*b*c^2*d^3*x^12 + 7155*a^2*c*d^4*x^12 + 7425*b^2*c^2*d^3*x^15 + 5940*a*b*c*d^4*x^
15 + 243*a^2*d^5*x^15 - 28*(c + d*x^3)^2*(27*b^2*c^2*x^6*(7*c + 6*d*x^3) + 9*a*b*c*x^3*(73*c^2 + 104*c*d*x^3 +
33*d^2*x^6) + a^2*(500*c^3 + 843*c^2*d*x^3 + 375*c*d^2*x^6 + 27*d^3*x^9))*Hypergeometric2F1[1/3, 1, 4/3, ((b*
c - a*d)*x^3)/(c*(a + b*x^3))]) - 486*(b*c - a*d)^4*x^12*(c + d*x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1
, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(16380*c^5*(b*c - a*d)^3*x^8*(a + b*x^3)^(7/3)*(c + d*x^3)^2)

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 )^{4/3} \left (c+d x^3\right )^3} \, 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 )^3} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac {65 c^2 \left (a+b x^3\right )^2 \left (14000 a^2 c^5+21896 a b c^5 x^3+48104 a^2 c^4 d x^3+8391 b^2 c^5 x^6+70802 a b c^4 d x^6+60807 a^2 c^3 d^2 x^6+24417 b^2 c^4 d x^9+81534 a b c^3 d^2 x^9+33657 a^2 c^2 d^3 x^9+23409 b^2 c^3 d^2 x^{12}+38652 a b c^2 d^3 x^{12}+7155 a^2 c d^4 x^{12}+7425 b^2 c^2 d^3 x^{15}+5940 a b c d^4 x^{15}+243 a^2 d^5 x^{15}-28 \left (c+d x^3\right )^2 \left (27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+a^2 \left (500 c^3+843 c^2 d x^3+375 c d^2 x^6+27 d^3 x^9\right )\right ) \, _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 )\right )-486 (b c-a d)^4 x^{12} \left (c+d x^3\right )^3 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{16380 c^5 (b c-a d)^3 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2}\\ \end {align*}

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Mathematica [C]  time = 3.08, size = 428, normalized size = 1.14 \[ -\frac {486 x^{12} \left (c+d x^3\right )^3 (b c-a d)^4 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )+65 c^2 \left (a+b x^3\right )^2 \left (28 \left (c+d x^3\right )^2 \left (a^2 \left (500 c^3+843 c^2 d x^3+375 c d^2 x^6+27 d^3 x^9\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-14000 a^2 c^5-48104 a^2 c^4 d x^3-60807 a^2 c^3 d^2 x^6-33657 a^2 c^2 d^3 x^9-7155 a^2 c d^4 x^{12}-243 a^2 d^5 x^{15}-21896 a b c^5 x^3-70802 a b c^4 d x^6-81534 a b c^3 d^2 x^9-38652 a b c^2 d^3 x^{12}-5940 a b c d^4 x^{15}-8391 b^2 c^5 x^6-24417 b^2 c^4 d x^9-23409 b^2 c^3 d^2 x^{12}-7425 b^2 c^2 d^3 x^{15}\right )}{16380 c^5 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/16380*(65*c^2*(a + b*x^3)^2*(-14000*a^2*c^5 - 21896*a*b*c^5*x^3 - 48104*a^2*c^4*d*x^3 - 8391*b^2*c^5*x^6 -
70802*a*b*c^4*d*x^6 - 60807*a^2*c^3*d^2*x^6 - 24417*b^2*c^4*d*x^9 - 81534*a*b*c^3*d^2*x^9 - 33657*a^2*c^2*d^3*
x^9 - 23409*b^2*c^3*d^2*x^12 - 38652*a*b*c^2*d^3*x^12 - 7155*a^2*c*d^4*x^12 - 7425*b^2*c^2*d^3*x^15 - 5940*a*b
*c*d^4*x^15 - 243*a^2*d^5*x^15 + 28*(c + d*x^3)^2*(27*b^2*c^2*x^6*(7*c + 6*d*x^3) + 9*a*b*c*x^3*(73*c^2 + 104*
c*d*x^3 + 33*d^2*x^6) + a^2*(500*c^3 + 843*c^2*d*x^3 + 375*c*d^2*x^6 + 27*d^3*x^9))*Hypergeometric2F1[1/3, 1,
4/3, ((b*c - a*d)*x^3)/(c*(a + b*x^3))]) + 486*(b*c - a*d)^4*x^12*(c + d*x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/
3}, {1, 1, 16/3}, ((b*c - a*d)*x^3)/(c*(a + b*x^3))])/(c^5*(-(b*c) + a*d)^3*x^8*(a + b*x^3)^(7/3)*(c + d*x^3)^
2)

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{4/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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)**(4/3)/(d*x**3+c)**3,x)

[Out]

Timed out

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