3.1686 \(\int (a+b x)^{5/4} (c+d x)^{5/4} \, dx\)

Optimal. Leaf size=408 \[ \frac{5 (b c-a d)^{9/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right ),\frac{1}{2}\right )}{168 \sqrt{2} b^{9/4} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}-\frac{5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{84 b^2 d^2}+\frac{(a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)^2}{42 b^2 d}+\frac{(a+b x)^{9/4} \sqrt [4]{c+d x} (b c-a d)}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b} \]

[Out]

(-5*(b*c - a*d)^3*(a + b*x)^(1/4)*(c + d*x)^(1/4))/(84*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/4)*(c + d*x)^(1/
4))/(42*b^2*d) + ((b*c - a*d)*(a + b*x)^(9/4)*(c + d*x)^(1/4))/(7*b^2) + (2*(a + b*x)^(9/4)*(c + d*x)^(5/4))/(
7*b) + (5*(b*c - a*d)^(9/2)*((a + b*x)*(c + d*x))^(3/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]*Sqrt[d]*
Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sq
rt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticF[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1
/4))/Sqrt[b*c - a*d]], 1/2])/(168*Sqrt[2]*b^(9/4)*d^(9/4)*(a + b*x)^(3/4)*(c + d*x)^(3/4)*(b*c + a*d + 2*b*d*x
)*Sqrt[(a*d + b*(c + 2*d*x))^2])

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Rubi [A]  time = 0.590231, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {50, 62, 623, 220} \[ -\frac{5 \sqrt [4]{a+b x} \sqrt [4]{c+d x} (b c-a d)^3}{84 b^2 d^2}+\frac{5 (b c-a d)^{9/2} ((a+b x) (c+d x))^{3/4} \sqrt{(a d+b c+2 b d x)^2} \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} b^{9/4} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (a d+b c+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}+\frac{(a+b x)^{5/4} \sqrt [4]{c+d x} (b c-a d)^2}{42 b^2 d}+\frac{(a+b x)^{9/4} \sqrt [4]{c+d x} (b c-a d)}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/4)*(c + d*x)^(5/4),x]

[Out]

(-5*(b*c - a*d)^3*(a + b*x)^(1/4)*(c + d*x)^(1/4))/(84*b^2*d^2) + ((b*c - a*d)^2*(a + b*x)^(5/4)*(c + d*x)^(1/
4))/(42*b^2*d) + ((b*c - a*d)*(a + b*x)^(9/4)*(c + d*x)^(1/4))/(7*b^2) + (2*(a + b*x)^(9/4)*(c + d*x)^(5/4))/(
7*b) + (5*(b*c - a*d)^(9/2)*((a + b*x)*(c + d*x))^(3/4)*Sqrt[(b*c + a*d + 2*b*d*x)^2]*(1 + (2*Sqrt[b]*Sqrt[d]*
Sqrt[(a + b*x)*(c + d*x)])/(b*c - a*d))*Sqrt[(a*d + b*(c + 2*d*x))^2/((b*c - a*d)^2*(1 + (2*Sqrt[b]*Sqrt[d]*Sq
rt[(a + b*x)*(c + d*x)])/(b*c - a*d))^2)]*EllipticF[2*ArcTan[(Sqrt[2]*b^(1/4)*d^(1/4)*((a + b*x)*(c + d*x))^(1
/4))/Sqrt[b*c - a*d]], 1/2])/(168*Sqrt[2]*b^(9/4)*d^(9/4)*(a + b*x)^(3/4)*(c + d*x)^(3/4)*(b*c + a*d + 2*b*d*x
)*Sqrt[(a*d + b*(c + 2*d*x))^2])

Rule 50

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 62

Int[((a_.) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Dist[((a + b*x)^m*(c + d*x)^m)/((a + b*x)
*(c + d*x))^m, Int[(a*c + (b*c + a*d)*x + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] &&
LtQ[-1, m, 0] && LeQ[3, Denominator[m], 4]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int (a+b x)^{5/4} (c+d x)^{5/4} \, dx &=\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{(5 (b c-a d)) \int (a+b x)^{5/4} \sqrt [4]{c+d x} \, dx}{14 b}\\ &=\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{(b c-a d)^2 \int \frac{(a+b x)^{5/4}}{(c+d x)^{3/4}} \, dx}{28 b^2}\\ &=\frac{(b c-a d)^2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{42 b^2 d}+\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}-\frac{\left (5 (b c-a d)^3\right ) \int \frac{\sqrt [4]{a+b x}}{(c+d x)^{3/4}} \, dx}{168 b^2 d}\\ &=-\frac{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{84 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{42 b^2 d}+\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{\left (5 (b c-a d)^4\right ) \int \frac{1}{(a+b x)^{3/4} (c+d x)^{3/4}} \, dx}{336 b^2 d^2}\\ &=-\frac{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{84 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{42 b^2 d}+\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{\left (5 (b c-a d)^4 ((a+b x) (c+d x))^{3/4}\right ) \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^{3/4}} \, dx}{336 b^2 d^2 (a+b x)^{3/4} (c+d x)^{3/4}}\\ &=-\frac{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{84 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{42 b^2 d}+\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{\left (5 (b c-a d)^4 ((a+b x) (c+d x))^{3/4} \sqrt{(b c+a d+2 b d x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-4 a b c d+(b c+a d)^2+4 b d x^4}} \, dx,x,\sqrt [4]{(a+b x) (c+d x)}\right )}{84 b^2 d^2 (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x)}\\ &=-\frac{5 (b c-a d)^3 \sqrt [4]{a+b x} \sqrt [4]{c+d x}}{84 b^2 d^2}+\frac{(b c-a d)^2 (a+b x)^{5/4} \sqrt [4]{c+d x}}{42 b^2 d}+\frac{(b c-a d) (a+b x)^{9/4} \sqrt [4]{c+d x}}{7 b^2}+\frac{2 (a+b x)^{9/4} (c+d x)^{5/4}}{7 b}+\frac{5 (b c-a d)^{9/2} ((a+b x) (c+d x))^{3/4} \sqrt{(b c+a d+2 b d x)^2} \left (1+\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}\right ) \sqrt{\frac{(a d+b (c+2 d x))^2}{(b c-a d)^2 \left (1+\frac{2 \sqrt{b} \sqrt{d} \sqrt{(a+b x) (c+d x)}}{b c-a d}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt [4]{d} \sqrt [4]{(a+b x) (c+d x)}}{\sqrt{b c-a d}}\right )|\frac{1}{2}\right )}{168 \sqrt{2} b^{9/4} d^{9/4} (a+b x)^{3/4} (c+d x)^{3/4} (b c+a d+2 b d x) \sqrt{(a d+b (c+2 d x))^2}}\\ \end{align*}

Mathematica [C]  time = 0.0583903, size = 73, normalized size = 0.18 \[ \frac{4 (a+b x)^{9/4} (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},\frac{9}{4};\frac{13}{4};\frac{d (a+b x)}{a d-b c}\right )}{9 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/4)*(c + d*x)^(5/4),x]

[Out]

(4*(a + b*x)^(9/4)*(c + d*x)^(5/4)*Hypergeometric2F1[-5/4, 9/4, 13/4, (d*(a + b*x))/(-(b*c) + a*d)])/(9*b*((b*
(c + d*x))/(b*c - a*d))^(5/4))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{5}{4}}} \left ( dx+c \right ) ^{{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/4)*(d*x+c)^(5/4),x)

[Out]

int((b*x+a)^(5/4)*(d*x+c)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{\frac{5}{4}}{\left (d x + c\right )}^{\frac{5}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/4)*(d*x+c)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(5/4)*(d*x + c)^(5/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/4)*(d*x+c)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [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((b*x+a)**(5/4)*(d*x+c)**(5/4),x)

[Out]

Timed out

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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((b*x+a)^(5/4)*(d*x+c)^(5/4),x, algorithm="giac")

[Out]

Timed out