3.18.36 \(\int (a+b x)^{5/2} \sqrt [6]{c+d x} \, dx\) [1736]

Optimal. Leaf size=487 \[ \frac {81 (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac {9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac {81\ 3^{3/4} (b c-a d)^{11/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2816 b d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \]

[Out]

-9/352*(-a*d+b*c)^2*(b*x+a)^(3/2)*(d*x+c)^(1/6)/b/d^2+3/176*(-a*d+b*c)*(b*x+a)^(5/2)*(d*x+c)^(1/6)/b/d+3/11*(b
*x+a)^(7/2)*(d*x+c)^(1/6)/b+81/1408*(-a*d+b*c)^3*(d*x+c)^(1/6)*(b*x+a)^(1/2)/b/d^3-81/2816*3^(3/4)*(-a*d+b*c)^
(11/3)*(d*x+c)^(1/6)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))*(((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1-3^(1
/2)))^2/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2)/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*
(1-3^(1/2)))*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))*EllipticF((1-((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+
c)^(1/3)*(1-3^(1/2)))^2/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))
*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b^(2/3)*(d*x+c)^(2/3))/((-a*d+b*c)^(1/3)-b^(1/3)*(d
*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2)/b/d^4/(b*x+a)^(1/2)/(-b^(1/3)*(d*x+c)^(1/3)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c
)^(1/3))/((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3)*(1+3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.38, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {52, 65, 231} \begin {gather*} -\frac {81\ 3^{3/4} \sqrt [6]{c+d x} (b c-a d)^{11/3} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\text {ArcCos}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2816 b d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {81 \sqrt {a+b x} \sqrt [6]{c+d x} (b c-a d)^3}{1408 b d^3}-\frac {9 (a+b x)^{3/2} \sqrt [6]{c+d x} (b c-a d)^2}{352 b d^2}+\frac {3 (a+b x)^{5/2} \sqrt [6]{c+d x} (b c-a d)}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2)*(c + d*x)^(1/6),x]

[Out]

(81*(b*c - a*d)^3*Sqrt[a + b*x]*(c + d*x)^(1/6))/(1408*b*d^3) - (9*(b*c - a*d)^2*(a + b*x)^(3/2)*(c + d*x)^(1/
6))/(352*b*d^2) + (3*(b*c - a*d)*(a + b*x)^(5/2)*(c + d*x)^(1/6))/(176*b*d) + (3*(a + b*x)^(7/2)*(c + d*x)^(1/
6))/(11*b) - (81*3^(3/4)*(b*c - a*d)^(11/3)*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt
[((b*c - a*d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3)
- (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d
*x)^(1/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(2816*b*d^4*Sqrt[a
+ b*x]*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1
+ Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)])

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 231

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s +
r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*(
(s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rubi steps

\begin {align*} \int (a+b x)^{5/2} \sqrt [6]{c+d x} \, dx &=\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}+\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{(c+d x)^{5/6}} \, dx}{22 b}\\ &=\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac {\left (15 (b c-a d)^2\right ) \int \frac {(a+b x)^{3/2}}{(c+d x)^{5/6}} \, dx}{352 b d}\\ &=-\frac {9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}+\frac {\left (27 (b c-a d)^3\right ) \int \frac {\sqrt {a+b x}}{(c+d x)^{5/6}} \, dx}{704 b d^2}\\ &=\frac {81 (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac {9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac {\left (81 (b c-a d)^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/6}} \, dx}{2816 b d^3}\\ &=\frac {81 (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac {9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac {\left (243 (b c-a d)^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^6}{d}}} \, dx,x,\sqrt [6]{c+d x}\right )}{1408 b d^4}\\ &=\frac {81 (b c-a d)^3 \sqrt {a+b x} \sqrt [6]{c+d x}}{1408 b d^3}-\frac {9 (b c-a d)^2 (a+b x)^{3/2} \sqrt [6]{c+d x}}{352 b d^2}+\frac {3 (b c-a d) (a+b x)^{5/2} \sqrt [6]{c+d x}}{176 b d}+\frac {3 (a+b x)^{7/2} \sqrt [6]{c+d x}}{11 b}-\frac {81\ 3^{3/4} (b c-a d)^{11/3} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2816 b d^4 \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.07, size = 73, normalized size = 0.15 \begin {gather*} \frac {2 (a+b x)^{7/2} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {1}{6},\frac {7}{2};\frac {9}{2};\frac {d (a+b x)}{-b c+a d}\right )}{7 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2)*(c + d*x)^(1/6),x]

[Out]

(2*(a + b*x)^(7/2)*(c + d*x)^(1/6)*Hypergeometric2F1[-1/6, 7/2, 9/2, (d*(a + b*x))/(-(b*c) + a*d)])/(7*b*((b*(
c + d*x))/(b*c - a*d))^(1/6))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {1}{6}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(d*x+c)^(1/6),x)

[Out]

int((b*x+a)^(5/2)*(d*x+c)^(1/6),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(d*x+c)^(1/6),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/2)*(d*x + c)^(1/6), x)

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Fricas [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((b*x+a)^(5/2)*(d*x+c)^(1/6),x, algorithm="fricas")

[Out]

integral((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*(d*x + c)^(1/6), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{\frac {5}{2}} \sqrt [6]{c + d x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(d*x+c)**(1/6),x)

[Out]

Integral((a + b*x)**(5/2)*(c + d*x)**(1/6), x)

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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((b*x+a)^(5/2)*(d*x+c)^(1/6),x, algorithm="giac")

[Out]

integrate((b*x + a)^(5/2)*(d*x + c)^(1/6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{1/6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/2)*(c + d*x)^(1/6),x)

[Out]

int((a + b*x)^(5/2)*(c + d*x)^(1/6), x)

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