Integrand size = 15, antiderivative size = 249 \[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\frac {224 a^6 x}{16575 b^4 \sqrt [4]{a x^2+b x^3}}-\frac {112 a^5 x^2}{49725 b^3 \sqrt [4]{a x^2+b x^3}}+\frac {56 a^4 x^3}{49725 b^2 \sqrt [4]{a x^2+b x^3}}-\frac {4 a^3 x^4}{5525 b \sqrt [4]{a x^2+b x^3}}+\frac {4 a^2 x^5}{425 \sqrt [4]{a x^2+b x^3}}+\frac {4}{75} a x^3 \left (a x^2+b x^3\right )^{3/4}+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}-\frac {448 a^{13/2} \sqrt {x} \sqrt [4]{\frac {a+b x}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |2\right )}{16575 b^{9/2} \sqrt [4]{a x^2+b x^3}} \] Output:
224/16575*a^6*x/b^4/(b*x^3+a*x^2)^(1/4)-112/49725*a^5*x^2/b^3/(b*x^3+a*x^2 )^(1/4)+56/49725*a^4*x^3/b^2/(b*x^3+a*x^2)^(1/4)-4/5525*a^3*x^4/b/(b*x^3+a *x^2)^(1/4)+4/425*a^2*x^5/(b*x^3+a*x^2)^(1/4)+4/75*a*x^3*(b*x^3+a*x^2)^(3/ 4)+4/25*x*(b*x^3+a*x^2)^(7/4)-448/16575*a^(13/2)*x^(1/2)*((b*x+a)/a)^(1/4) *EllipticE(sin(1/2*arctan(b^(1/2)*x^(1/2)/a^(1/2))),2^(1/2))/b^(9/2)/(b*x^ 3+a*x^2)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.20 \[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\frac {2 a x^3 \left (x^2 (a+b x)\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {9}{2},\frac {11}{2},-\frac {b x}{a}\right )}{9 \left (1+\frac {b x}{a}\right )^{3/4}} \] Input:
Integrate[(a*x^2 + b*x^3)^(7/4),x]
Output:
(2*a*x^3*(x^2*(a + b*x))^(3/4)*Hypergeometric2F1[-7/4, 9/2, 11/2, -((b*x)/ a)])/(9*(1 + (b*x)/a)^(3/4))
Time = 1.12 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {1910, 1927, 1930, 1930, 1930, 1930, 1917, 73, 836, 27, 765, 762, 1390, 1389, 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.
| \(\displaystyle \int \left (a x^2+b x^3\right )^{7/4} \, dx\) |
| \(\Big \downarrow \) 1910 |
| \(\displaystyle \frac {7}{25} a \int x^2 \left (b x^3+a x^2\right )^{3/4}dx+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \int \frac {x^4}{\sqrt [4]{b x^3+a x^2}}dx+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \int \frac {x^3}{\sqrt [4]{b x^3+a x^2}}dx}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \int \frac {x^2}{\sqrt [4]{b x^3+a x^2}}dx}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \int \frac {x}{\sqrt [4]{b x^3+a x^2}}dx}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {2 a \int \frac {1}{\sqrt [4]{b x^3+a x^2}}dx}{5 b}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {2 a \sqrt {x} \sqrt [4]{a+b x} \int \frac {1}{\sqrt {x} \sqrt [4]{a+b x}}dx}{5 b \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \int \frac {\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 836 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\sqrt {a} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {a} \sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\sqrt {a} \int \frac {1}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {1}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}d\sqrt [4]{a+b x}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\frac {\sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {a}+\sqrt {a+b x}}{\sqrt {1-\frac {a+b x}{a}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\frac {\sqrt {a} \sqrt {1-\frac {a+b x}{a}} \int \frac {\sqrt {\frac {\sqrt {a+b x}}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {a+b x}}{\sqrt {a}}}}d\sqrt [4]{a+b x}}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {7}{25} a \left (\frac {1}{7} a \left (\frac {4 x^2 \left (a x^2+b x^3\right )^{3/4}}{17 b}-\frac {14 a \left (\frac {4 x \left (a x^2+b x^3\right )^{3/4}}{13 b}-\frac {10 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{9 b}-\frac {2 a \left (\frac {4 \left (a x^2+b x^3\right )^{3/4}}{5 b x}-\frac {8 a \sqrt {x} \sqrt [4]{a+b x} \left (\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}-\frac {a^{3/4} \sqrt {1-\frac {a+b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{a+b x}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {\frac {a+b x}{b}-\frac {a}{b}}}\right )}{5 b^2 \sqrt [4]{a x^2+b x^3}}\right )}{3 b}\right )}{13 b}\right )}{17 b}\right )+\frac {4}{21} x^3 \left (a x^2+b x^3\right )^{3/4}\right )+\frac {4}{25} x \left (a x^2+b x^3\right )^{7/4}\) |
Input:
Int[(a*x^2 + b*x^3)^(7/4),x]
Output:
(4*x*(a*x^2 + b*x^3)^(7/4))/25 + (7*a*((4*x^3*(a*x^2 + b*x^3)^(3/4))/21 + (a*((4*x^2*(a*x^2 + b*x^3)^(3/4))/(17*b) - (14*a*((4*x*(a*x^2 + b*x^3)^(3/ 4))/(13*b) - (10*a*((4*(a*x^2 + b*x^3)^(3/4))/(9*b) - (2*a*((4*(a*x^2 + b* x^3)^(3/4))/(5*b*x) - (8*a*Sqrt[x]*(a + b*x)^(1/4)*((a^(3/4)*Sqrt[1 - (a + b*x)/a]*EllipticE[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b] - (a^(3/4)*Sqrt[1 - (a + b*x)/a]*EllipticF[ArcSin[(a + b*x)^(1/4 )/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b]))/(5*b^2*(a*x^2 + b*x^3)^(1/4) )))/(3*b)))/(13*b)))/(17*b)))/7))/25
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
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])
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Simp[a*(n - j)*(p/(n*p + 1)) Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]) Int[ x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && !Integ erQ[p] && NeQ[n, j] && PosQ[n - j]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
\[\int \left (b \,x^{3}+a \,x^{2}\right )^{\frac {7}{4}}d x\]
Input:
int((b*x^3+a*x^2)^(7/4),x)
Output:
int((b*x^3+a*x^2)^(7/4),x)
\[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{\frac {7}{4}} \,d x } \] Input:
integrate((b*x^3+a*x^2)^(7/4),x, algorithm="fricas")
Output:
integral((b*x^3 + a*x^2)^(7/4), x)
\[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\int \left (a x^{2} + b x^{3}\right )^{\frac {7}{4}}\, dx \] Input:
integrate((b*x**3+a*x**2)**(7/4),x)
Output:
Integral((a*x**2 + b*x**3)**(7/4), x)
\[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{\frac {7}{4}} \,d x } \] Input:
integrate((b*x^3+a*x^2)^(7/4),x, algorithm="maxima")
Output:
integrate((b*x^3 + a*x^2)^(7/4), x)
\[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\int { {\left (b x^{3} + a x^{2}\right )}^{\frac {7}{4}} \,d x } \] Input:
integrate((b*x^3+a*x^2)^(7/4),x, algorithm="giac")
Output:
integrate((b*x^3 + a*x^2)^(7/4), x)
Time = 9.39 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15 \[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\frac {2\,x\,{\left (b\,x^3+a\,x^2\right )}^{7/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {9}{2};\ \frac {11}{2};\ -\frac {b\,x}{a}\right )}{9\,{\left (\frac {b\,x}{a}+1\right )}^{7/4}} \] Input:
int((a*x^2 + b*x^3)^(7/4),x)
Output:
(2*x*(a*x^2 + b*x^3)^(7/4)*hypergeom([-7/4, 9/2], 11/2, -(b*x)/a))/(9*((b* x)/a + 1)^(7/4))
\[ \int \left (a x^2+b x^3\right )^{7/4} \, dx=\frac {-\frac {224 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a^{5}}{16575}+\frac {112 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a^{4} b x}{9945}-\frac {56 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a^{3} b^{2} x^{2}}{5525}+\frac {4 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a^{2} b^{3} x^{3}}{425}+\frac {16 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} a \,b^{4} x^{4}}{75}+\frac {4 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}} b^{5} x^{5}}{25}+\frac {112 \left (\int \frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}}}{b \,x^{2}+a x}d x \right ) a^{6}}{16575}}{b^{4}} \] Input:
int((b*x^3+a*x^2)^(7/4),x)
Output:
(4*( - 168*sqrt(x)*(a + b*x)**(3/4)*a**5 + 140*sqrt(x)*(a + b*x)**(3/4)*a* *4*b*x - 126*sqrt(x)*(a + b*x)**(3/4)*a**3*b**2*x**2 + 117*sqrt(x)*(a + b* x)**(3/4)*a**2*b**3*x**3 + 2652*sqrt(x)*(a + b*x)**(3/4)*a*b**4*x**4 + 198 9*sqrt(x)*(a + b*x)**(3/4)*b**5*x**5 + 84*int((sqrt(x)*(a + b*x)**(3/4))/( a*x + b*x**2),x)*a**6))/(49725*b**4)