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]

-(d*x)/(6*c*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3)^2) + (b*(6*b*c + a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^3
)^(1/3)*(c + d*x^3)) + (d*(18*b^2*c^2 + 15*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/(18*a*c^2*(b*c - a*d)^3*(
c + d*x^3)) - (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3
)^(1/3)))/Sqrt[3]])/(9*Sqrt[3]*c^(8/3)*(b*c - a*d)^(10/3)) - (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[c +
d*x^3])/(54*c^(8/3)*(b*c - a*d)^(10/3)) + (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[((b*c - a*d)^(1/3)*x)/c
^(1/3) - (a + b*x^3)^(1/3)])/(18*c^(8/3)*(b*c - a*d)^(10/3))

________________________________________________________________________________________

Rubi [C]  time = 2.74231, 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 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 )^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*}

Mathematica [C]  time = 2.26589, size = 428, normalized size = 1.14 \[ -\frac{486 x^{12} \left (c+d x^3\right )^3 (b c-a d)^4 \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{16}{3}\right \},\frac{x^3 (b c-a d)}{c \left (a+b x^3\right )}\right )+65 c^2 \left (a+b x^3\right )^2 \left (28 \left (c+d x^3\right )^2 \left (a^2 \left (843 c^2 d x^3+500 c^3+375 c d^2 x^6+27 d^3 x^9\right )+9 a b c x^3 \left (73 c^2+104 c d x^3+33 d^2 x^6\right )+27 b^2 c^2 x^6 \left (7 c+6 d x^3\right )\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(b c-a d) x^3}{c \left (b x^3+a\right )}\right )-33657 a^2 c^2 d^3 x^9-60807 a^2 c^3 d^2 x^6-48104 a^2 c^4 d x^3-14000 a^2 c^5-7155 a^2 c d^4 x^{12}-243 a^2 d^5 x^{15}-38652 a b c^2 d^3 x^{12}-81534 a b c^3 d^2 x^9-70802 a b c^4 d x^6-21896 a b c^5 x^3-5940 a b c d^4 x^{15}-7425 b^2 c^2 d^3 x^{15}-23409 b^2 c^3 d^2 x^{12}-24417 b^2 c^4 d x^9-8391 b^2 c^5 x^6\right )}{16380 c^5 x^8 \left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^2 (a d-b c)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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 - 23
409*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)

________________________________________________________________________________________

Maple [F]  time = 0.435, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \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)^3,x)

[Out]

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

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**3,x)

[Out]

Timed out

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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