3.1678 \(\int (a+b x)^{3/4} (c+d x)^{5/4} \, dx\)

Optimal. Leaf size=205 \[ \frac{5 (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac{5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \]

[Out]

(5*(b*c - a*d)^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(96*b^2*d) + (5*(b*c - a*d)*(a + b*x)^(7/4)*(c + d*x)^(1/4))
/(24*b^2) + ((a + b*x)^(7/4)*(c + d*x)^(5/4))/(3*b) + (5*(b*c - a*d)^3*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/
4)*(c + d*x)^(1/4))])/(64*b^(9/4)*d^(7/4)) - (5*(b*c - a*d)^3*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c +
d*x)^(1/4))])/(64*b^(9/4)*d^(7/4))

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Rubi [A]  time = 0.13922, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {50, 63, 331, 298, 205, 208} \[ \frac{5 (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}+\frac{5 (a+b x)^{3/4} \sqrt [4]{c+d x} (b c-a d)^2}{96 b^2 d}+\frac{5 (a+b x)^{7/4} \sqrt [4]{c+d x} (b c-a d)}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]

[Out]

(5*(b*c - a*d)^2*(a + b*x)^(3/4)*(c + d*x)^(1/4))/(96*b^2*d) + (5*(b*c - a*d)*(a + b*x)^(7/4)*(c + d*x)^(1/4))
/(24*b^2) + ((a + b*x)^(7/4)*(c + d*x)^(5/4))/(3*b) + (5*(b*c - a*d)^3*ArcTan[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/
4)*(c + d*x)^(1/4))])/(64*b^(9/4)*d^(7/4)) - (5*(b*c - a*d)^3*ArcTanh[(d^(1/4)*(a + b*x)^(1/4))/(b^(1/4)*(c +
d*x)^(1/4))])/(64*b^(9/4)*d^(7/4))

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 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 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 298

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

Rubi steps

\begin{align*} \int (a+b x)^{3/4} (c+d x)^{5/4} \, dx &=\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac{(5 (b c-a d)) \int (a+b x)^{3/4} \sqrt [4]{c+d x} \, dx}{12 b}\\ &=\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac{\left (5 (b c-a d)^2\right ) \int \frac{(a+b x)^{3/4}}{(c+d x)^{3/4}} \, dx}{96 b^2}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac{\left (5 (b c-a d)^3\right ) \int \frac{1}{\sqrt [4]{a+b x} (c+d x)^{3/4}} \, dx}{128 b^2 d}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac{\left (5 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (c-\frac{a d}{b}+\frac{d x^4}{b}\right )^{3/4}} \, dx,x,\sqrt [4]{a+b x}\right )}{32 b^3 d}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac{\left (5 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{d x^4}{b}} \, dx,x,\frac{\sqrt [4]{a+b x}}{\sqrt [4]{c+d x}}\right )}{32 b^3 d}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}-\frac{\left (5 (b c-a d)^3\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 )}{64 b^2 d^{3/2}}+\frac{\left (5 (b c-a d)^3\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 )}{64 b^2 d^{3/2}}\\ &=\frac{5 (b c-a d)^2 (a+b x)^{3/4} \sqrt [4]{c+d x}}{96 b^2 d}+\frac{5 (b c-a d) (a+b x)^{7/4} \sqrt [4]{c+d x}}{24 b^2}+\frac{(a+b x)^{7/4} (c+d x)^{5/4}}{3 b}+\frac{5 (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}-\frac{5 (b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt [4]{d} \sqrt [4]{a+b x}}{\sqrt [4]{b} \sqrt [4]{c+d x}}\right )}{64 b^{9/4} d^{7/4}}\\ \end{align*}

Mathematica [C]  time = 0.0599047, size = 73, normalized size = 0.36 \[ \frac{4 (a+b x)^{7/4} (c+d x)^{5/4} \, _2F_1\left (-\frac{5}{4},\frac{7}{4};\frac{11}{4};\frac{d (a+b x)}{a d-b c}\right )}{7 b \left (\frac{b (c+d x)}{b c-a d}\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/4)*(c + d*x)^(5/4),x]

[Out]

(4*(a + b*x)^(7/4)*(c + d*x)^(5/4)*Hypergeometric2F1[-5/4, 7/4, 11/4, (d*(a + b*x))/(-(b*c) + a*d)])/(7*b*((b*
(c + d*x))/(b*c - a*d))^(5/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{{\frac{3}{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)^(3/4)*(d*x+c)^(5/4),x)

[Out]

int((b*x+a)^(3/4)*(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{3}{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)^(3/4)*(d*x+c)^(5/4),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(3/4)*(d*x + c)^(5/4), x)

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Fricas [B]  time = 4.42256, size = 4761, normalized size = 23.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/4)*(d*x+c)^(5/4),x, algorithm="fricas")

[Out]

-1/384*(60*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8
*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3
*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4)*arctan(((b^10*c^3*d^5 - 3*a*b^9*c
^2*d^6 + 3*a^2*b^8*c*d^7 - a^3*b^7*d^8)*(b*x + a)^(3/4)*(d*x + c)^(1/4)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^
2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*
a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d
^12)/(b^9*d^7))^(3/4) + (b^8*d^5*x + a*b^7*d^5)*sqrt(((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b
^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(b*x + a)*sqrt(d*x + c) + (b^5*d^4*x + a*b^4*d^
4)*sqrt((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792
*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*
a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7)))/(b*x + a))*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a
^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792
*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*
d^12)/(b^9*d^7))^(3/4))/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a
^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a
^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12 + (b^13*c^12 - 12*a*b^12*c^11*d + 66*a^2
*b^11*c^10*d^2 - 220*a^3*b^10*c^9*d^3 + 495*a^4*b^9*c^8*d^4 - 792*a^5*b^8*c^7*d^5 + 924*a^6*b^7*c^6*d^6 - 792*
a^7*b^6*c^5*d^7 + 495*a^8*b^5*c^4*d^8 - 220*a^9*b^4*c^3*d^9 + 66*a^10*b^3*c^2*d^10 - 12*a^11*b^2*c*d^11 + a^12
*b*d^12)*x)) + 15*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*
b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*
b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4)*log(-5*((b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/4)*(d*x + c)^(1/4) + (b^3*d^2*x + a*b^2*d^2)*((b^12*c^12 - 12*a
*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^
6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^
11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4))/(b*x + a)) - 15*b^2*d*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*
c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5
*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b
^9*d^7))^(1/4)*log(-5*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x + a)^(3/4)*(d*x + c)^(1/4) - (
b^3*d^2*x + a*b^2*d^2)*((b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b
^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b
^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(b^9*d^7))^(1/4))/(b*x + a)) - 4*(32*b^2*d^2
*x^2 + 5*b^2*c^2 + 42*a*b*c*d - 15*a^2*d^2 + 4*(13*b^2*c*d + 3*a*b*d^2)*x)*(b*x + a)^(3/4)*(d*x + c)^(1/4))/(b
^2*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right )^{\frac{3}{4}} \left (c + d x\right )^{\frac{5}{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/4)*(d*x+c)**(5/4),x)

[Out]

Integral((a + b*x)**(3/4)*(c + d*x)**(5/4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{\frac{3}{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)^(3/4)*(d*x+c)^(5/4),x, algorithm="giac")

[Out]

integrate((b*x + a)^(3/4)*(d*x + c)^(5/4), x)