3.74 \(\int \frac {(c+d x^3)^2}{(a+b x^3)^{7/3}} \, dx\)

Optimal. Leaf size=152 \[ \frac {x (b c-a d) (4 a d+3 b c)}{4 a^2 b^2 \sqrt [3]{a+b x^3}}-\frac {d^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{7/3}}+\frac {d^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}+\frac {x \left (c+d x^3\right ) (b c-a d)}{4 a b \left (a+b x^3\right )^{4/3}} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {413, 385, 239} \[ \frac {x (b c-a d) (4 a d+3 b c)}{4 a^2 b^2 \sqrt [3]{a+b x^3}}-\frac {d^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 b^{7/3}}+\frac {d^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}+\frac {x \left (c+d x^3\right ) (b c-a d)}{4 a b \left (a+b x^3\right )^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^2/(a + b*x^3)^(7/3),x]

[Out]

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 385

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

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{7/3}} \, dx &=\frac {(b c-a d) x \left (c+d x^3\right )}{4 a b \left (a+b x^3\right )^{4/3}}+\frac {\int \frac {c (3 b c+a d)+4 a d^2 x^3}{\left (a+b x^3\right )^{4/3}} \, dx}{4 a b}\\ &=\frac {(b c-a d) (3 b c+4 a d) x}{4 a^2 b^2 \sqrt [3]{a+b x^3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{4 a b \left (a+b x^3\right )^{4/3}}+\frac {d^2 \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{b^2}\\ &=\frac {(b c-a d) (3 b c+4 a d) x}{4 a^2 b^2 \sqrt [3]{a+b x^3}}+\frac {(b c-a d) x \left (c+d x^3\right )}{4 a b \left (a+b x^3\right )^{4/3}}+\frac {d^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3}}-\frac {d^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 b^{7/3}}\\ \end {align*}

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Mathematica [A]  time = 5.25, size = 180, normalized size = 1.18 \[ \frac {x \left (\left (a+b x^3\right ) \left (-5 a^2 d^2+2 a b c d+3 b^2 c^2\right )+a (b c-a d)^2\right )}{4 a^2 b^2 \left (a+b x^3\right )^{4/3}}+\frac {d^2 \left (\log \left (\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )\right )}{6 b^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^2/(a + b*x^3)^(7/3),x]

[Out]

(x*(a*(b*c - a*d)^2 + (3*b^2*c^2 + 2*a*b*c*d - 5*a^2*d^2)*(a + b*x^3)))/(4*a^2*b^2*(a + b*x^3)^(4/3)) + (d^2*(
2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + L
og[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(6*b^(7/3))

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fricas [B]  time = 0.74, size = 719, normalized size = 4.73 \[ \left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{3} d^{2} x^{6} + 2 \, a^{3} b^{2} d^{2} x^{3} + a^{4} b d^{2}\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) - 4 \, {\left (a^{2} b^{2} d^{2} x^{6} + 2 \, a^{3} b d^{2} x^{3} + a^{4} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 2 \, {\left (a^{2} b^{2} d^{2} x^{6} + 2 \, a^{3} b d^{2} x^{3} + a^{4} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left ({\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{4} + 4 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{12 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a^{2} b^{3} d^{2} x^{6} + 2 \, a^{3} b^{2} d^{2} x^{3} + a^{4} b d^{2}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 4 \, {\left (a^{2} b^{2} d^{2} x^{6} + 2 \, a^{3} b d^{2} x^{3} + a^{4} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 2 \, {\left (a^{2} b^{2} d^{2} x^{6} + 2 \, a^{3} b d^{2} x^{3} + a^{4} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left ({\left (3 \, b^{4} c^{2} + 2 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{4} + 4 \, {\left (a b^{3} c^{2} - a^{3} b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{12 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(7/3),x, algorithm="fricas")

[Out]

[1/12*(6*sqrt(1/3)*(a^2*b^3*d^2*x^6 + 2*a^3*b^2*d^2*x^3 + a^4*b*d^2)*sqrt((-b)^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3
+ a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*((-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b
)^(2/3)*x)*sqrt((-b)^(1/3)/b) + 2*a) - 4*(a^2*b^2*d^2*x^6 + 2*a^3*b*d^2*x^3 + a^4*d^2)*(-b)^(2/3)*log(((-b)^(1
/3)*x + (b*x^3 + a)^(1/3))/x) + 2*(a^2*b^2*d^2*x^6 + 2*a^3*b*d^2*x^3 + a^4*d^2)*(-b)^(2/3)*log(((-b)^(2/3)*x^2
- (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) + 3*((3*b^4*c^2 + 2*a*b^3*c*d - 5*a^2*b^2*d^2)*x^4
+ 4*(a*b^3*c^2 - a^3*b*d^2)*x)*(b*x^3 + a)^(2/3))/(a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3), -1/12*(12*sqrt(1/3
)*(a^2*b^3*d^2*x^6 + 2*a^3*b^2*d^2*x^3 + a^4*b*d^2)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b
*x^3 + a)^(1/3))*sqrt(-(-b)^(1/3)/b)/x) + 4*(a^2*b^2*d^2*x^6 + 2*a^3*b*d^2*x^3 + a^4*d^2)*(-b)^(2/3)*log(((-b)
^(1/3)*x + (b*x^3 + a)^(1/3))/x) - 2*(a^2*b^2*d^2*x^6 + 2*a^3*b*d^2*x^3 + a^4*d^2)*(-b)^(2/3)*log(((-b)^(2/3)*
x^2 - (b*x^3 + a)^(1/3)*(-b)^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) - 3*((3*b^4*c^2 + 2*a*b^3*c*d - 5*a^2*b^2*d^2)*
x^4 + 4*(a*b^3*c^2 - a^3*b*d^2)*x)*(b*x^3 + a)^(2/3))/(a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(7/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^2/(b*x^3 + a)^(7/3), x)

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maple [F]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{3}+c \right )^{2}}{\left (b \,x^{3}+a \right )^{\frac {7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^2/(b*x^3+a)^(7/3),x)

[Out]

int((d*x^3+c)^2/(b*x^3+a)^(7/3),x)

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maxima [A]  time = 1.19, size = 190, normalized size = 1.25 \[ -\frac {{\left (b - \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} c^{2} x^{4}}{4 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}} + \frac {c d x^{4}}{2 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a} - \frac {1}{12} \, {\left (\frac {3 \, {\left (b + \frac {4 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} x^{4}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}} + \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} - \frac {2 \, \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} d^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^2/(b*x^3+a)^(7/3),x, algorithm="maxima")

[Out]

-1/4*(b - 4*(b*x^3 + a)/x^3)*c^2*x^4/((b*x^3 + a)^(4/3)*a^2) + 1/2*c*d*x^4/((b*x^3 + a)^(4/3)*a) - 1/12*(3*(b
+ 4*(b*x^3 + a)/x^3)*x^4/((b*x^3 + a)^(4/3)*b^2) + 4*sqrt(3)*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)
/x)/b^(1/3))/b^(7/3) - 2*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(7/3) + 4*log(-b
^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(7/3))*d^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{7/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^2/(a + b*x^3)^(7/3),x)

[Out]

int((c + d*x^3)^2/(a + b*x^3)^(7/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**2/(b*x**3+a)**(7/3),x)

[Out]

Integral((c + d*x**3)**2/(a + b*x**3)**(7/3), x)

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