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

Optimal. Leaf size=130 \[ -\frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac {(3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \]

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Rubi [A]  time = 0.08, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {80, 63, 240, 212, 208, 205} \begin {gather*} -\frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac {(3 a d+b c) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac {\sqrt [4]{a+b x} (c+d x)^{3/4}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1/4)*(c + d*x)^(3/4))/(b*d) - ((b*c + 3*a*d)*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(
1/4))])/(2*b^(7/4)*d^(5/4)) - ((b*c + 3*a*d)*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*
b^(7/4)*d^(5/4))

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 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 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]

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 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 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]

Rubi steps

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

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Mathematica [C]  time = 0.10, size = 91, normalized size = 0.70 \begin {gather*} \frac {\sqrt [4]{a+b x} \left (b (c+d x)-(3 a d+b c) \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 )\right )}{b^2 d \sqrt [4]{c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.39, size = 160, normalized size = 1.23 \begin {gather*} \frac {(-3 a d-b c) \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}+\frac {(-3 a d-b c) \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{2 b^{7/4} d^{5/4}}-\frac {\sqrt [4]{a+b x} (a d-b c)}{b d \sqrt [4]{c+d x} \left (b-\frac {d (a+b x)}{c+d x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((-(b*c) + a*d)*(a + b*x)^(1/4))/(b*d*(c + d*x)^(1/4)*(b - (d*(a + b*x))/(c + d*x)))) + ((-(b*c) - 3*a*d)*Ar
cTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4)) + ((-(b*c) - 3*a*d)*ArcTanh[(d^
(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c + d*x)^(1/4))])/(2*b^(7/4)*d^(5/4))

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fricas [B]  time = 1.59, size = 839, normalized size = 6.45 \begin {gather*} \frac {4 \, b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b^{6} c d^{4} + 3 \, a b^{5} d^{5}\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {3}{4}} - {\left (b^{5} d^{5} x + b^{5} c d^{4}\right )} \sqrt {\frac {{\left (b^{2} c^{2} + 6 \, a b c d + 9 \, a^{2} d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} + {\left (b^{4} d^{3} x + b^{4} c d^{2}\right )} \sqrt {\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}}}{d x + c}} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {3}{4}}}{b^{4} c^{5} + 12 \, a b^{3} c^{4} d + 54 \, a^{2} b^{2} c^{3} d^{2} + 108 \, a^{3} b c^{2} d^{3} + 81 \, a^{4} c d^{4} + {\left (b^{4} c^{4} d + 12 \, a b^{3} c^{3} d^{2} + 54 \, a^{2} b^{2} c^{2} d^{3} + 108 \, a^{3} b c d^{4} + 81 \, a^{4} d^{5}\right )} x}\right ) - b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} + {\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + b d \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b c + 3 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}} - {\left (b^{2} d^{2} x + b^{2} c d\right )} \left (\frac {b^{4} c^{4} + 12 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 108 \, a^{3} b c d^{3} + 81 \, a^{4} d^{4}}{b^{7} d^{5}}\right )^{\frac {1}{4}}}{d x + c}\right ) + 4 \, {\left (b x + a\right )}^{\frac {1}{4}} {\left (d x + c\right )}^{\frac {3}{4}}}{4 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*ar
ctan(-((b^6*c*d^4 + 3*a*b^5*d^5)*(b*x + a)^(1/4)*(d*x + c)^(3/4)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d
^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(3/4) - (b^5*d^5*x + b^5*c*d^4)*sqrt(((b^2*c^2 + 6*a*b*c*d + 9*a
^2*d^2)*sqrt(b*x + a)*sqrt(d*x + c) + (b^4*d^3*x + b^4*c*d^2)*sqrt((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*
d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5)))/(d*x + c))*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 1
08*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(3/4))/(b^4*c^5 + 12*a*b^3*c^4*d + 54*a^2*b^2*c^3*d^2 + 108*a^3*b*c^2*
d^3 + 81*a^4*c*d^4 + (b^4*c^4*d + 12*a*b^3*c^3*d^2 + 54*a^2*b^2*c^2*d^3 + 108*a^3*b*c*d^4 + 81*a^4*d^5)*x)) -
b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log(((b*c
+ 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) + (b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*
d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4))/(d*x + c)) + b*d*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2
*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^5))^(1/4)*log(((b*c + 3*a*d)*(b*x + a)^(1/4)*(d*x + c)^(3/4) -
(b^2*d^2*x + b^2*c*d)*((b^4*c^4 + 12*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 108*a^3*b*c*d^3 + 81*a^4*d^4)/(b^7*d^
5))^(1/4))/(d*x + c)) + 4*(b*x + a)^(1/4)*(d*x + c)^(3/4))/(b*d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (b x +a \right )^{\frac {3}{4}} \left (d x +c \right )^{\frac {1}{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{{\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (a + b x\right )^{\frac {3}{4}} \sqrt [4]{c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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