3.17.35 \(\int (a+b x)^{3/2} (c+d x)^{3/4} \, dx\) [1635]

3.17.35.1 Optimal result

Integrand size = 19, antiderivative size = 270 \[ \int (a+b x)^{3/2} (c+d x)^{3/4} \, dx=-\frac {8 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/4}}{65 b d^2}+\frac {4 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/4}}{39 b d}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}+\frac {16 (b c-a d)^{15/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{65 b^{7/4} d^3 \sqrt {a+b x}}-\frac {16 (b c-a d)^{15/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{65 b^{7/4} d^3 \sqrt {a+b x}} \]

output

4/39*(-a*d+b*c)*(b*x+a)^(3/2)*(d*x+c)^(3/4)/b/d+4/13*(b*x+a)^(5/2)*(d*x+c) 
^(3/4)/b-8/65*(-a*d+b*c)^2*(d*x+c)^(3/4)*(b*x+a)^(1/2)/b/d^2+16/65*(-a*d+b 
*c)^(15/4)*EllipticE(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a) 
/(-a*d+b*c))^(1/2)/b^(7/4)/d^3/(b*x+a)^(1/2)-16/65*(-a*d+b*c)^(15/4)*Ellip 
ticF(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)*(-d*(b*x+a)/(-a*d+b*c))^(1/ 
2)/b^(7/4)/d^3/(b*x+a)^(1/2)
 

3.17.35.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.27 \[ \int (a+b x)^{3/2} (c+d x)^{3/4} \, dx=\frac {2 (a+b x)^{5/2} (c+d x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {5}{2},\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \left (\frac {b (c+d x)}{b c-a d}\right )^{3/4}} \]

input

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

output

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

3.17.35.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {60, 60, 60, 73, 836, 765, 762, 1390, 1388, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^{3/2} (c+d x)^{3/4} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt [4]{c+d x}}dx}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt [4]{c+d x}}dx}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {2 (b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}}dx}{5 d}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{\sqrt {b}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 1388

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {\sqrt {b c-a d} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {\sqrt {\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}+1}}{\sqrt {1-\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}}}d\sqrt [4]{c+d x}}{\sqrt {b} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {3 (b c-a d) \left (\frac {4 (a+b x)^{3/2} (c+d x)^{3/4}}{9 d}-\frac {2 (b c-a d) \left (\frac {4 \sqrt {a+b x} (c+d x)^{3/4}}{5 d}-\frac {8 (b c-a d) \left (\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {(b c-a d)^{3/4} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{5 d^2}\right )}{3 d}\right )}{13 b}+\frac {4 (a+b x)^{5/2} (c+d x)^{3/4}}{13 b}\)

input

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

output

(4*(a + b*x)^(5/2)*(c + d*x)^(3/4))/(13*b) + (3*(b*c - a*d)*((4*(a + b*x)^ 
(3/2)*(c + d*x)^(3/4))/(9*d) - (2*(b*c - a*d)*((4*Sqrt[a + b*x]*(c + d*x)^ 
(3/4))/(5*d) - (8*(b*c - a*d)*(((b*c - a*d)^(3/4)*Sqrt[1 - (b*(c + d*x))/( 
b*c - a*d)]*EllipticE[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - a*d)^(1/4)], 
 -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) - ((b*c - a*d)^(3/4)*S 
qrt[1 - (b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/ 
4))/(b*c - a*d)^(1/4)], -1])/(b^(3/4)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]) 
))/(5*d^2)))/(3*d)))/(13*b)
 

3.17.35.3.1 Defintions of rubi rules used

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 836

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

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

3.17.35.4 Maple [F]

\[\int \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{4}}d x\]

input

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

output

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

3.17.35.5 Fricas [F]

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

input

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

output

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

3.17.35.6 Sympy [F]

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

input

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

output

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

3.17.35.7 Maxima [F]

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

input

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

output

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

3.17.35.8 Giac [F]

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

input

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

output

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

3.17.35.9 Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{3/2} (c+d x)^{3/4} \, dx=\int {\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/4} \,d x \]

input

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

output

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