3.2979 \(\int (d x)^m \sqrt {a+b \sqrt {c x^3}} \, dx\)

Optimal. Leaf size=84 \[ \frac {x (d x)^m \sqrt {a+b \sqrt {c x^3}} \, _2F_1\left (-\frac {1}{2},\frac {2 (m+1)}{3};\frac {1}{3} (2 m+5);-\frac {b \sqrt {c x^3}}{a}\right )}{(m+1) \sqrt {\frac {b \sqrt {c x^3}}{a}+1}} \]

[Out]

x*(d*x)^m*hypergeom([-1/2, 2/3+2/3*m],[5/3+2/3*m],-b*(c*x^3)^(1/2)/a)*(a+b*(c*x^3)^(1/2))^(1/2)/(1+m)/(1+b*(c*
x^3)^(1/2)/a)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {369, 343, 341, 365, 364} \[ \frac {x (d x)^m \sqrt {a+b \sqrt {c x^3}} \, _2F_1\left (-\frac {1}{2},\frac {2 (m+1)}{3};\frac {1}{3} (2 m+5);-\frac {b \sqrt {c x^3}}{a}\right )}{(m+1) \sqrt {\frac {b \sqrt {c x^3}}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(x*(d*x)^m*Sqrt[a + b*Sqrt[c*x^3]]*Hypergeometric2F1[-1/2, (2*(1 + m))/3, (5 + 2*m)/3, -((b*Sqrt[c*x^3])/a)])/
((1 + m)*Sqrt[1 + (b*Sqrt[c*x^3])/a])

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 343

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rubi steps

\begin {align*} \int (d x)^m \sqrt {a+b \sqrt {c x^3}} \, dx &=\operatorname {Subst}\left (\int (d x)^m \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\left (x^{-m} (d x)^m\right ) \int x^m \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname {Subst}\left (\int x^{-1+2 (1+m)} \sqrt {a+b \sqrt {c} x^3} \, dx,x,\sqrt {x}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {\left (2 x^{-m} (d x)^m \sqrt {a+b \sqrt {c} x^{3/2}}\right ) \operatorname {Subst}\left (\int x^{-1+2 (1+m)} \sqrt {1+\frac {b \sqrt {c} x^3}{a}} \, dx,x,\sqrt {x}\right )}{\sqrt {1+\frac {b \sqrt {c} x^{3/2}}{a}}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {x (d x)^m \sqrt {a+b \sqrt {c x^3}} \, _2F_1\left (-\frac {1}{2},\frac {2 (1+m)}{3};\frac {1}{3} (5+2 m);-\frac {b \sqrt {c x^3}}{a}\right )}{(1+m) \sqrt {1+\frac {b \sqrt {c x^3}}{a}}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 84, normalized size = 1.00 \[ \frac {x (d x)^m \sqrt {a+b \sqrt {c x^3}} \, _2F_1\left (-\frac {1}{2},\frac {2 (m+1)}{3};\frac {1}{3} (2 m+5);-\frac {b \sqrt {c x^3}}{a}\right )}{(m+1) \sqrt {\frac {b \sqrt {c x^3}}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(x*(d*x)^m*Sqrt[a + b*Sqrt[c*x^3]]*Hypergeometric2F1[-1/2, (2*(1 + m))/3, (5 + 2*m)/3, -((b*Sqrt[c*x^3])/a)])/
((1 + m)*Sqrt[1 + (b*Sqrt[c*x^3])/a])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   alglogextint: unimplemented

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*(d*x)^m, x)

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\sqrt {c \,x^{3}}\, b}\, \left (d x \right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a+(c*x^3)^(1/2)*b)^(1/2),x)

[Out]

int((d*x)^m*(a+(c*x^3)^(1/2)*b)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(c*x^3)*b + a)*(d*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*(c*x^3)^(1/2))^(1/2),x)

[Out]

int((d*x)^m*(a + b*(c*x^3)^(1/2))^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*(c*x**3)**(1/2))**(1/2),x)

[Out]

Integral((d*x)**m*sqrt(a + b*sqrt(c*x**3)), x)

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