3.1413 \(\int \frac {(a+b x+c x^2)^{4/3}}{(b d+2 c d x)^{17/3}} \, dx\)

Optimal. Leaf size=44 \[ \frac {3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {682} \[ \frac {3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x]

[Out]

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

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{17/3}} \, dx &=\frac {3 \left (a+b x+c x^2\right )^{7/3}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{14/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 50, normalized size = 1.14 \[ \frac {3 (a+x (b+c x))^{7/3} \sqrt [3]{d (b+2 c x)}}{7 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.36, size = 194, normalized size = 4.41 \[ \frac {3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} {\left (2 \, c d x + b d\right )}^{\frac {1}{3}} {\left (c x^{2} + b x + a\right )}^{\frac {1}{3}}}{7 \, {\left (32 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \, {\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \, {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \, {\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x + {\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(17/3),x, algorithm="fricas")

[Out]

3/7*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*(2*c*d*x + b*d)^(1/3)*(c*x^2 + b*x + a)^(1/3)/(3
2*(b^2*c^5 - 4*a*c^6)*d^6*x^5 + 80*(b^3*c^4 - 4*a*b*c^5)*d^6*x^4 + 80*(b^4*c^3 - 4*a*b^2*c^4)*d^6*x^3 + 40*(b^
5*c^2 - 4*a*b^3*c^3)*d^6*x^2 + 10*(b^6*c - 4*a*b^4*c^2)*d^6*x + (b^7 - 4*a*b^5*c)*d^6)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {17}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(17/3),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(17/3), x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 44, normalized size = 1.00 \[ -\frac {3 \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}{7 \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {17}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(17/3),x)

[Out]

-3/7*(2*c*x+b)*(c*x^2+b*x+a)^(7/3)/(4*a*c-b^2)/(2*c*d*x+b*d)^(17/3)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + b x + a\right )}^{\frac {4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac {17}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(4/3)/(2*c*d*x+b*d)^(17/3),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(4/3)/(2*c*d*x + b*d)^(17/3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{4/3}}{{\left (b\,d+2\,c\,d\,x\right )}^{17/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3),x)

[Out]

int((a + b*x + c*x^2)^(4/3)/(b*d + 2*c*d*x)^(17/3), x)

________________________________________________________________________________________

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((c*x**2+b*x+a)**(4/3)/(2*c*d*x+b*d)**(17/3),x)

[Out]

Timed out

________________________________________________________________________________________