3.1.61 \(\int \frac {c+d x^3}{(a+b x^3)^{10/3}} \, dx\) [61]

Optimal. Leaf size=91 \[ \frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) x}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {3 (6 b c+a d) x}{28 a^3 b \sqrt [3]{a+b x^3}} \]

[Out]

1/7*(-a*d+b*c)*x/a/b/(b*x^3+a)^(7/3)+1/28*(a*d+6*b*c)*x/a^2/b/(b*x^3+a)^(4/3)+3/28*(a*d+6*b*c)*x/a^3/b/(b*x^3+
a)^(1/3)

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Rubi [A]
time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {393, 198, 197} \begin {gather*} \frac {3 x (a d+6 b c)}{28 a^3 b \sqrt [3]{a+b x^3}}+\frac {x (a d+6 b c)}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {x (b c-a d)}{7 a b \left (a+b x^3\right )^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)/(a + b*x^3)^(10/3),x]

[Out]

((b*c - a*d)*x)/(7*a*b*(a + b*x^3)^(7/3)) + ((6*b*c + a*d)*x)/(28*a^2*b*(a + b*x^3)^(4/3)) + (3*(6*b*c + a*d)*
x)/(28*a^3*b*(a + b*x^3)^(1/3))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

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

Rule 393

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])

Rubi steps

\begin {align*} \int \frac {c+d x^3}{\left (a+b x^3\right )^{10/3}} \, dx &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{7 a b}\\ &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) x}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {(3 (6 b c+a d)) \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{28 a^2 b}\\ &=\frac {(b c-a d) x}{7 a b \left (a+b x^3\right )^{7/3}}+\frac {(6 b c+a d) x}{28 a^2 b \left (a+b x^3\right )^{4/3}}+\frac {3 (6 b c+a d) x}{28 a^3 b \sqrt [3]{a+b x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 60, normalized size = 0.66 \begin {gather*} \frac {28 a^2 c x+42 a b c x^4+7 a^2 d x^4+18 b^2 c x^7+3 a b d x^7}{28 a^3 \left (a+b x^3\right )^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)/(a + b*x^3)^(10/3),x]

[Out]

(28*a^2*c*x + 42*a*b*c*x^4 + 7*a^2*d*x^4 + 18*b^2*c*x^7 + 3*a*b*d*x^7)/(28*a^3*(a + b*x^3)^(7/3))

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Maple [A]
time = 0.25, size = 57, normalized size = 0.63

method	result	size
gosper	\(\frac {x \left (3 a b d \,x^{6}+18 b^{2} c \,x^{6}+7 a^{2} d \,x^{3}+42 a b c \,x^{3}+28 a^{2} c \right )}{28 \left (b \,x^{3}+a \right )^{\frac {7}{3}} a^{3}}\)	\(57\)
trager	\(\frac {x \left (3 a b d \,x^{6}+18 b^{2} c \,x^{6}+7 a^{2} d \,x^{3}+42 a b c \,x^{3}+28 a^{2} c \right )}{28 \left (b \,x^{3}+a \right )^{\frac {7}{3}} a^{3}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)/(b*x^3+a)^(10/3),x,method=_RETURNVERBOSE)

[Out]

1/28*x*(3*a*b*d*x^6+18*b^2*c*x^6+7*a^2*d*x^3+42*a*b*c*x^3+28*a^2*c)/(b*x^3+a)^(7/3)/a^3

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Maxima [A]
time = 0.27, size = 86, normalized size = 0.95 \begin {gather*} -\frac {{\left (4 \, b - \frac {7 \, {\left (b x^{3} + a\right )}}{x^{3}}\right )} d x^{7}}{28 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{2}} + \frac {{\left (2 \, b^{2} - \frac {7 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {14 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} c x^{7}}{14 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(10/3),x, algorithm="maxima")

[Out]

-1/28*(4*b - 7*(b*x^3 + a)/x^3)*d*x^7/((b*x^3 + a)^(7/3)*a^2) + 1/14*(2*b^2 - 7*(b*x^3 + a)*b/x^3 + 14*(b*x^3
+ a)^2/x^6)*c*x^7/((b*x^3 + a)^(7/3)*a^3)

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Fricas [A]
time = 6.96, size = 87, normalized size = 0.96 \begin {gather*} \frac {{\left (3 \, {\left (6 \, b^{2} c + a b d\right )} x^{7} + 7 \, {\left (6 \, a b c + a^{2} d\right )} x^{4} + 28 \, a^{2} c x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{28 \, {\left (a^{3} b^{3} x^{9} + 3 \, a^{4} b^{2} x^{6} + 3 \, a^{5} b x^{3} + a^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(10/3),x, algorithm="fricas")

[Out]

1/28*(3*(6*b^2*c + a*b*d)*x^7 + 7*(6*a*b*c + a^2*d)*x^4 + 28*a^2*c*x)*(b*x^3 + a)^(2/3)/(a^3*b^3*x^9 + 3*a^4*b
^2*x^6 + 3*a^5*b*x^3 + a^6)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (83) = 166\).
time = 111.05, size = 709, normalized size = 7.79 \begin {gather*} c \left (\frac {28 a^{5} x \Gamma \left (\frac {1}{3}\right )}{27 a^{\frac {25}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {22}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {19}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 27 a^{\frac {16}{3}} b^{3} x^{9} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )} + \frac {70 a^{4} b x^{4} \Gamma \left (\frac {1}{3}\right )}{27 a^{\frac {25}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {22}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {19}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 27 a^{\frac {16}{3}} b^{3} x^{9} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )} + \frac {60 a^{3} b^{2} x^{7} \Gamma \left (\frac {1}{3}\right )}{27 a^{\frac {25}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {22}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {19}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 27 a^{\frac {16}{3}} b^{3} x^{9} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )} + \frac {18 a^{2} b^{3} x^{10} \Gamma \left (\frac {1}{3}\right )}{27 a^{\frac {25}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {22}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 81 a^{\frac {19}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 27 a^{\frac {16}{3}} b^{3} x^{9} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )}\right ) + d \left (\frac {7 a x^{4} \Gamma \left (\frac {4}{3}\right )}{9 a^{\frac {13}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 18 a^{\frac {10}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 9 a^{\frac {7}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )} + \frac {3 b x^{7} \Gamma \left (\frac {4}{3}\right )}{9 a^{\frac {13}{3}} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 18 a^{\frac {10}{3}} b x^{3} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right ) + 9 a^{\frac {7}{3}} b^{2} x^{6} \sqrt [3]{1 + \frac {b x^{3}}{a}} \Gamma \left (\frac {10}{3}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)/(b*x**3+a)**(10/3),x)

[Out]

c*(28*a**5*x*gamma(1/3)/(27*a**(25/3)*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 81*a**(22/3)*b*x**3*(1 + b*x**3/a)**
(1/3)*gamma(10/3) + 81*a**(19/3)*b**2*x**6*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 27*a**(16/3)*b**3*x**9*(1 + b*x
**3/a)**(1/3)*gamma(10/3)) + 70*a**4*b*x**4*gamma(1/3)/(27*a**(25/3)*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 81*a*
*(22/3)*b*x**3*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 81*a**(19/3)*b**2*x**6*(1 + b*x**3/a)**(1/3)*gamma(10/3) +
27*a**(16/3)*b**3*x**9*(1 + b*x**3/a)**(1/3)*gamma(10/3)) + 60*a**3*b**2*x**7*gamma(1/3)/(27*a**(25/3)*(1 + b*
x**3/a)**(1/3)*gamma(10/3) + 81*a**(22/3)*b*x**3*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 81*a**(19/3)*b**2*x**6*(1
+ b*x**3/a)**(1/3)*gamma(10/3) + 27*a**(16/3)*b**3*x**9*(1 + b*x**3/a)**(1/3)*gamma(10/3)) + 18*a**2*b**3*x**
10*gamma(1/3)/(27*a**(25/3)*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 81*a**(22/3)*b*x**3*(1 + b*x**3/a)**(1/3)*gamm
a(10/3) + 81*a**(19/3)*b**2*x**6*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 27*a**(16/3)*b**3*x**9*(1 + b*x**3/a)**(1
/3)*gamma(10/3))) + d*(7*a*x**4*gamma(4/3)/(9*a**(13/3)*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 18*a**(10/3)*b*x**
3*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 9*a**(7/3)*b**2*x**6*(1 + b*x**3/a)**(1/3)*gamma(10/3)) + 3*b*x**7*gamma
(4/3)/(9*a**(13/3)*(1 + b*x**3/a)**(1/3)*gamma(10/3) + 18*a**(10/3)*b*x**3*(1 + b*x**3/a)**(1/3)*gamma(10/3) +
9*a**(7/3)*b**2*x**6*(1 + b*x**3/a)**(1/3)*gamma(10/3)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)/(b*x^3+a)^(10/3),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)/(b*x^3 + a)^(10/3), x)

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Mupad [B]
time = 1.42, size = 87, normalized size = 0.96 \begin {gather*} \frac {3\,a\,d\,x\,{\left (b\,x^3+a\right )}^2-4\,a^3\,d\,x+18\,b\,c\,x\,{\left (b\,x^3+a\right )}^2+a^2\,d\,x\,\left (b\,x^3+a\right )+4\,a^2\,b\,c\,x+6\,a\,b\,c\,x\,\left (b\,x^3+a\right )}{28\,a^3\,b\,{\left (b\,x^3+a\right )}^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)/(a + b*x^3)^(10/3),x)

[Out]

(3*a*d*x*(a + b*x^3)^2 - 4*a^3*d*x + 18*b*c*x*(a + b*x^3)^2 + a^2*d*x*(a + b*x^3) + 4*a^2*b*c*x + 6*a*b*c*x*(a
+ b*x^3))/(28*a^3*b*(a + b*x^3)^(7/3))

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