3.901 \(\int \frac{x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx\)

Optimal. Leaf size=201 \[ \frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \]

[Out]

-((5*b*c + 7*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*b^2*d^2) + (x*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(2*b*d) +
((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(11/
4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4)
)])/(16*b^(11/4)*d^(9/4))

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Rubi [A]  time = 0.171786, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {90, 80, 63, 240, 212, 208, 205} \[ \frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}-\frac{\sqrt [4]{a+b x} (c+d x)^{3/4} (7 a d+5 b c)}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

-((5*b*c + 7*a*d)*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(8*b^2*d^2) + (x*(a + b*x)^(1/4)*(c + d*x)^(3/4))/(2*b*d) +
((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(16*b^(11/
4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4)
)])/(16*b^(11/4)*d^(9/4))

Rule 90

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx &=\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\int \frac{-a c-\frac{1}{4} (5 b c+7 a d) x}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{2 b d}\\ &=-\frac{(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{1}{(a+b x)^{3/4} \sqrt [4]{c+d x}} \, dx}{32 b^2 d^2}\\ &=-\frac{(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{c-\frac{a d}{b}+\frac{d x^4}{b}}} \, dx,x,\sqrt [4]{a+b x}\right )}{8 b^3 d^2}\\ &=-\frac{(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{8 b^3 d^2}\\ &=-\frac{(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}-\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 b^{5/2} d^2}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b}+\sqrt{d} x^2} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{16 b^{5/2} d^2}\\ &=-\frac{(5 b c+7 a d) \sqrt [4]{a+b x} (c+d x)^{3/4}}{8 b^2 d^2}+\frac{x \sqrt [4]{a+b x} (c+d x)^{3/4}}{2 b d}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{16 b^{11/4} d^{9/4}}\\ \end{align*}

Mathematica [C]  time = 0.124692, size = 123, normalized size = 0.61 \[ \frac{\sqrt [4]{a+b x} \left (\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \sqrt [4]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{d (a+b x)}{a d-b c}\right )-b (c+d x) (7 a d+5 b c-4 b d x)\right )}{8 b^3 d^2 \sqrt [4]{c+d x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((a + b*x)^(3/4)*(c + d*x)^(1/4)),x]

[Out]

((a + b*x)^(1/4)*(-(b*(c + d*x)*(5*b*c + 7*a*d - 4*b*d*x)) + (5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*((b*(c + d*x
))/(b*c - a*d))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (d*(a + b*x))/(-(b*c) + a*d)]))/(8*b^3*d^2*(c + d*x)^(1
/4))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( bx+a \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

[Out]

int(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x + a\right )}^{\frac{3}{4}}{\left (d x + c\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="maxima")

[Out]

integrate(x^2/((b*x + a)^(3/4)*(d*x + c)^(1/4)), x)

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Fricas [B]  time = 2.56695, size = 3661, normalized size = 18.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="fricas")

[Out]

-1/32*(4*b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4
*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^
9))^(1/4)*arctan(-((5*b^10*c^2*d^7 + 6*a*b^9*c*d^8 + 21*a^2*b^8*d^9)*(b*x + a)^(1/4)*(d*x + c)^(3/4)*((625*b^8
*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*
b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(3/4) - (b^8*d^8*x + b
^8*c*d^7)*sqrt(((25*b^4*c^4 + 60*a*b^3*c^3*d + 246*a^2*b^2*c^2*d^2 + 252*a^3*b*c*d^3 + 441*a^4*d^4)*sqrt(b*x +
a)*sqrt(d*x + c) + (b^6*d^5*x + b^6*c*d^4)*sqrt((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 421
20*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c
*d^7 + 194481*a^8*d^8)/(b^11*d^9)))/(d*x + c))*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 4212
0*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*
d^7 + 194481*a^8*d^8)/(b^11*d^9))^(3/4))/(625*b^8*c^9 + 3000*a*b^7*c^8*d + 15900*a^2*b^6*c^7*d^2 + 42120*a^3*b
^5*c^6*d^3 + 112806*a^4*b^4*c^5*d^4 + 176904*a^5*b^3*c^4*d^5 + 280476*a^6*b^2*c^3*d^6 + 222264*a^7*b*c^2*d^7 +
194481*a^8*c*d^8 + (625*b^8*c^8*d + 3000*a*b^7*c^7*d^2 + 15900*a^2*b^6*c^6*d^3 + 42120*a^3*b^5*c^5*d^4 + 1128
06*a^4*b^4*c^4*d^5 + 176904*a^5*b^3*c^3*d^6 + 280476*a^6*b^2*c^2*d^7 + 222264*a^7*b*c*d^8 + 194481*a^8*d^9)*x)
) - b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*
c^4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(
1/4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^3*d^3*x + b^3*c*d^2)*((625
*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*
a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4))/(d*x + c))
+ b^2*d^2*((625*b^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^
4*d^4 + 176904*a^5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/
4)*log(((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*(b*x + a)^(1/4)*(d*x + c)^(3/4) - (b^3*d^3*x + b^3*c*d^2)*((625*b
^8*c^8 + 3000*a*b^7*c^7*d + 15900*a^2*b^6*c^6*d^2 + 42120*a^3*b^5*c^5*d^3 + 112806*a^4*b^4*c^4*d^4 + 176904*a^
5*b^3*c^3*d^5 + 280476*a^6*b^2*c^2*d^6 + 222264*a^7*b*c*d^7 + 194481*a^8*d^8)/(b^11*d^9))^(1/4))/(d*x + c)) -
4*(4*b*d*x - 5*b*c - 7*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(b^2*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b x\right )^{\frac{3}{4}} \sqrt [4]{c + d x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x+a)**(3/4)/(d*x+c)**(1/4),x)

[Out]

Integral(x**2/((a + b*x)**(3/4)*(c + d*x)**(1/4)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(b*x+a)^(3/4)/(d*x+c)^(1/4),x, algorithm="giac")

[Out]

Timed out