3.132 \(\int \frac {(c+d x^3)^{23/12}}{(a+b x^3)^{13/4}} \, dx\)

Optimal. Leaf size=153 \[ \frac {253 c x \left (c+d x^3\right )^{11/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac {1}{3},\frac {5}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {92 c x \left (c+d x^3\right )^{11/12}}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {4 x \left (c+d x^3\right )^{23/12}}{27 a \left (a+b x^3\right )^{9/4}} \]

[Out]

92/405*c*x*(d*x^3+c)^(11/12)/a^2/(b*x^3+a)^(5/4)+4/27*x*(d*x^3+c)^(23/12)/a/(b*x^3+a)^(9/4)+253/405*c*x*(c*(b*
x^3+a)/a/(d*x^3+c))^(5/4)*(d*x^3+c)^(11/12)*hypergeom([1/3, 5/4],[4/3],-(-a*d+b*c)*x^3/a/(d*x^3+c))/a^2/(b*x^3
+a)^(5/4)

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Rubi [A]  time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {378, 380} \[ \frac {253 c x \left (c+d x^3\right )^{11/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac {1}{3},\frac {5}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {92 c x \left (c+d x^3\right )^{11/12}}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {4 x \left (c+d x^3\right )^{23/12}}{27 a \left (a+b x^3\right )^{9/4}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^(23/12)/(a + b*x^3)^(13/4),x]

[Out]

(92*c*x*(c + d*x^3)^(11/12))/(405*a^2*(a + b*x^3)^(5/4)) + (4*x*(c + d*x^3)^(23/12))/(27*a*(a + b*x^3)^(9/4))
+ (253*c*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 5/4, 4/3, -(((b*
c - a*d)*x^3)/(a*(c + d*x^3)))])/(405*a^2*(a + b*x^3)^(5/4))

Rule 378

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

Rule 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^p*Hypergeome
tric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d
*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^3\right )^{23/12}}{\left (a+b x^3\right )^{13/4}} \, dx &=\frac {4 x \left (c+d x^3\right )^{23/12}}{27 a \left (a+b x^3\right )^{9/4}}+\frac {(23 c) \int \frac {\left (c+d x^3\right )^{11/12}}{\left (a+b x^3\right )^{9/4}} \, dx}{27 a}\\ &=\frac {92 c x \left (c+d x^3\right )^{11/12}}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {4 x \left (c+d x^3\right )^{23/12}}{27 a \left (a+b x^3\right )^{9/4}}+\frac {\left (253 c^2\right ) \int \frac {1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx}{405 a^2}\\ &=\frac {92 c x \left (c+d x^3\right )^{11/12}}{405 a^2 \left (a+b x^3\right )^{5/4}}+\frac {4 x \left (c+d x^3\right )^{23/12}}{27 a \left (a+b x^3\right )^{9/4}}+\frac {253 c x \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \left (c+d x^3\right )^{11/12} \, _2F_1\left (\frac {1}{3},\frac {5}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{405 a^2 \left (a+b x^3\right )^{5/4}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 90, normalized size = 0.59 \[ \frac {c x \sqrt [4]{\frac {b x^3}{a}+1} \left (c+d x^3\right )^{11/12} \, _2F_1\left (\frac {1}{3},\frac {13}{4};\frac {4}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{a^3 \sqrt [4]{a+b x^3} \left (\frac {d x^3}{c}+1\right )^{5/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x^3)^(23/12)/(a + b*x^3)^(13/4),x]

[Out]

(c*x*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 13/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c +
d*x^3))])/(a^3*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(5/4))

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fricas [F]  time = 66.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}} {\left (d x^{3} + c\right )}^{\frac {23}{12}}}{b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(23/12)/(b*x^3+a)^(13/4),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(3/4)*(d*x^3 + c)^(23/12)/(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4), x
)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {23}{12}}}{{\left (b x^{3} + a\right )}^{\frac {13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(23/12)/(b*x^3+a)^(13/4),x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^(23/12)/(b*x^3 + a)^(13/4), x)

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maple [F]  time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{3}+c \right )^{\frac {23}{12}}}{\left (b \,x^{3}+a \right )^{\frac {13}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^(23/12)/(b*x^3+a)^(13/4),x)

[Out]

int((d*x^3+c)^(23/12)/(b*x^3+a)^(13/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {23}{12}}}{{\left (b x^{3} + a\right )}^{\frac {13}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^(23/12)/(b*x^3+a)^(13/4),x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)^(23/12)/(b*x^3 + a)^(13/4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,x^3+c\right )}^{23/12}}{{\left (b\,x^3+a\right )}^{13/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3)^(23/12)/(a + b*x^3)^(13/4),x)

[Out]

int((c + d*x^3)^(23/12)/(a + b*x^3)^(13/4), 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((d*x**3+c)**(23/12)/(b*x**3+a)**(13/4),x)

[Out]

Timed out

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