Integrand size = 31, antiderivative size = 215 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^4 (d+e x)^3}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (e f-9 d g) (e f+d g)^3 (d+e x)^2}{15 d^2 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 (e f+d g)^2 \left (e^2 f^2-8 d e f g+36 d^2 g^2\right ) (d+e x)}{15 d^3 e^5 \sqrt {d^2-e^2 x^2}}+\frac {g^4 \sqrt {d^2-e^2 x^2}}{e^5}-\frac {g^3 (4 e f+3 d g) \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5} \] Output:
1/5*(d*g+e*f)^4*(e*x+d)^3/d/e^5/(-e^2*x^2+d^2)^(5/2)+2/15*(-9*d*g+e*f)*(d* g+e*f)^3*(e*x+d)^2/d^2/e^5/(-e^2*x^2+d^2)^(3/2)+2/15*(d*g+e*f)^2*(36*d^2*g ^2-8*d*e*f*g+e^2*f^2)*(e*x+d)/d^3/e^5/(-e^2*x^2+d^2)^(1/2)+g^4*(-e^2*x^2+d ^2)^(1/2)/e^5-g^3*(3*d*g+4*e*f)*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^5
Time = 1.59 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (72 d^6 g^4+2 e^6 f^4 x^2+d^5 e g^3 (88 f-171 g x)-6 d e^5 f^3 x (f+2 g x)+3 d^4 e^2 g^2 \left (4 f^2-68 f g x+39 g^2 x^2\right )+d^2 e^4 f^2 \left (7 f^2+36 f g x+42 g^2 x^2\right )-d^3 e^3 g \left (12 f^3+36 f^2 g x-128 f g^2 x^2+15 g^3 x^3\right )\right )}{15 d^3 e^5 (d-e x)^3}+\frac {g^3 (4 e f+3 d g) \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^4 \sqrt {-e^2}} \] Input:
Integrate[((d + e*x)^3*(f + g*x)^4)/(d^2 - e^2*x^2)^(7/2),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(72*d^6*g^4 + 2*e^6*f^4*x^2 + d^5*e*g^3*(88*f - 171*g *x) - 6*d*e^5*f^3*x*(f + 2*g*x) + 3*d^4*e^2*g^2*(4*f^2 - 68*f*g*x + 39*g^2 *x^2) + d^2*e^4*f^2*(7*f^2 + 36*f*g*x + 42*g^2*x^2) - d^3*e^3*g*(12*f^3 + 36*f^2*g*x - 128*f*g^2*x^2 + 15*g^3*x^3)))/(15*d^3*e^5*(d - e*x)^3) + (g^3 *(4*e*f + 3*d*g)*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(e^4*Sqrt[-e^ 2])
Time = 1.01 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {691, 25, 2166, 25, 2166, 27, 455, 224, 216}
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.
\(\displaystyle \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 691 |
\(\displaystyle \frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {(d+e x)^2 \left (-\frac {5 d x^3 g^4}{e}-\frac {5 d (4 e f+d g) x^2 g^3}{e^2}-\frac {5 d \left (6 e^2 f^2+4 d e g f+d^2 g^2\right ) x g^2}{e^3}+\frac {2 e^4 f^4-12 d e^3 g f^3-18 d^2 e^2 g^2 f^2-12 d^3 e g^3 f-3 d^4 g^4}{e^4}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (-\frac {5 d x^3 g^4}{e}-\frac {5 d (4 e f+d g) x^2 g^3}{e^2}-\frac {5 d \left (6 e^2 f^2+4 d e g f+d^2 g^2\right ) x g^2}{e^3}+\frac {2 e^4 f^4-12 d e^3 g f^3-18 d^2 e^2 g^2 f^2-12 d^3 e g^3 f-3 d^4 g^4}{e^4}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 2166 |
\(\displaystyle \frac {\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {(d+e x) \left (\frac {15 d^2 x^2 g^4}{e^2}+\frac {30 d^2 (2 e f+d g) x g^3}{e^3}+\frac {2 e^4 f^4-12 d e^3 g f^3+42 d^2 e^2 g^2 f^2+68 d^3 e g^3 f+27 d^4 g^4}{e^4}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (\frac {15 d^2 x^2 g^4}{e^2}+\frac {30 d^2 (2 e f+d g) x g^3}{e^3}+\frac {2 e^4 f^4-12 d e^3 g f^3+42 d^2 e^2 g^2 f^2+68 d^3 e g^3 f+27 d^4 g^4}{e^4}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 2166 |
\(\displaystyle \frac {\frac {\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{d e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^3 g^3 (4 e f+3 d g+e g x)}{e^4 \sqrt {d^2-e^2 x^2}}dx}{d}}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{d e^5 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 g^3 \int \frac {4 e f+3 d g+e g x}{\sqrt {d^2-e^2 x^2}}dx}{e^4}}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{d e^5 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 g^3 \left ((3 d g+4 e f) \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {g \sqrt {d^2-e^2 x^2}}{e}\right )}{e^4}}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{d e^5 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 g^3 \left ((3 d g+4 e f) \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {g \sqrt {d^2-e^2 x^2}}{e}\right )}{e^4}}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {\frac {2 (d+e x) (d g+e f)^2 \left (36 d^2 g^2-8 d e f g+e^2 f^2\right )}{d e^5 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 g^3 \left (\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) (3 d g+4 e f)}{e}-\frac {g \sqrt {d^2-e^2 x^2}}{e}\right )}{e^4}}{3 d}+\frac {2 (d+e x)^2 (e f-9 d g) (d g+e f)^3}{3 d e^5 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^4}{5 d e^5 \left (d^2-e^2 x^2\right )^{5/2}}\) |
Input:
Int[((d + e*x)^3*(f + g*x)^4)/(d^2 - e^2*x^2)^(7/2),x]
Output:
((e*f + d*g)^4*(d + e*x)^3)/(5*d*e^5*(d^2 - e^2*x^2)^(5/2)) + ((2*(e*f - 9 *d*g)*(e*f + d*g)^3*(d + e*x)^2)/(3*d*e^5*(d^2 - e^2*x^2)^(3/2)) + ((2*(e* f + d*g)^2*(e^2*f^2 - 8*d*e*f*g + 36*d^2*g^2)*(d + e*x))/(d*e^5*Sqrt[d^2 - e^2*x^2]) - (15*d^2*g^3*(-((g*Sqrt[d^2 - e^2*x^2])/e) + ((4*e*f + 3*d*g)* ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/e^4)/(3*d))/(5*d)
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d* x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[(-d)*R*( d + e*x)^m*((a + c*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + R*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 + a*e^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(480\) vs. \(2(199)=398\).
Time = 1.36 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.24
method | result | size |
risch | \(\frac {g^{4} \sqrt {-e^{2} x^{2}+d^{2}}}{e^{5}}-\frac {\frac {\left (3 d g +4 e f \right ) g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e \sqrt {e^{2}}}+\frac {\left (g^{4} d^{4}+4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}+4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d e \left (x -\frac {d}{e}\right )^{3}}-\frac {2 e \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{5 d}\right )}{e^{4}}+\frac {4 g \left (d^{3} g^{3}+3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{e^{3}}+\frac {6 g^{2} \left (d^{2} g^{2}+2 d e f g +e^{2} f^{2}\right ) \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{3} d \left (x -\frac {d}{e}\right )}}{e^{3}}\) | \(481\) |
default | \(d^{3} f^{4} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+e^{2} g^{3} \left (3 d g +4 e f \right ) \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+\frac {d^{2} f^{3} \left (4 d g +3 e f \right )}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+3 g^{2} e \left (d^{2} g^{2}+4 d e f g +2 e^{2} f^{2}\right ) \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+3 d \,f^{2} \left (2 d^{2} g^{2}+4 d e f g +e^{2} f^{2}\right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+g \left (d^{3} g^{3}+12 d^{2} e f \,g^{2}+18 d \,e^{2} f^{2} g +4 e^{3} f^{3}\right ) \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+f \left (4 d^{3} g^{3}+18 d^{2} e f \,g^{2}+12 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+e^{3} g^{4} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )\) | \(822\) |
Input:
int((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
g^4*(-e^2*x^2+d^2)^(1/2)/e^5-1/e^3*((3*d*g+4*e*f)*g^3/e/(e^2)^(1/2)*arctan ((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/e^4*(d^4*g^4+4*d^3*e*f*g^3+6*d^2*e^ 2*f^2*g^2+4*d*e^3*f^3*g+e^4*f^4)*(1/5/d/e/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e* (x-d/e))^(1/2)-2/5*e/d*(1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^( 1/2)-1/3/d^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)))+4/e^3*g*(d^3*g ^3+3*d^2*e*f*g^2+3*d*e^2*f^2*g+e^3*f^3)*(1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2 -2*d*e*(x-d/e))^(1/2)-1/3/d^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2) )+6*g^2/e^3*(d^2*g^2+2*d*e*f*g+e^2*f^2)/d/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x -d/e))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (199) = 398\).
Time = 0.18 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.90 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {7 \, d^{3} e^{4} f^{4} - 12 \, d^{4} e^{3} f^{3} g + 12 \, d^{5} e^{2} f^{2} g^{2} + 88 \, d^{6} e f g^{3} + 72 \, d^{7} g^{4} - {\left (7 \, e^{7} f^{4} - 12 \, d e^{6} f^{3} g + 12 \, d^{2} e^{5} f^{2} g^{2} + 88 \, d^{3} e^{4} f g^{3} + 72 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (7 \, d e^{6} f^{4} - 12 \, d^{2} e^{5} f^{3} g + 12 \, d^{3} e^{4} f^{2} g^{2} + 88 \, d^{4} e^{3} f g^{3} + 72 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (7 \, d^{2} e^{5} f^{4} - 12 \, d^{3} e^{4} f^{3} g + 12 \, d^{4} e^{3} f^{2} g^{2} + 88 \, d^{5} e^{2} f g^{3} + 72 \, d^{6} e g^{4}\right )} x + 30 \, {\left (4 \, d^{6} e f g^{3} + 3 \, d^{7} g^{4} - {\left (4 \, d^{3} e^{4} f g^{3} + 3 \, d^{4} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (4 \, d^{4} e^{3} f g^{3} + 3 \, d^{5} e^{2} g^{4}\right )} x^{2} - 3 \, {\left (4 \, d^{5} e^{2} f g^{3} + 3 \, d^{6} e g^{4}\right )} x\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, d^{3} e^{3} g^{4} x^{3} - 7 \, d^{2} e^{4} f^{4} + 12 \, d^{3} e^{3} f^{3} g - 12 \, d^{4} e^{2} f^{2} g^{2} - 88 \, d^{5} e f g^{3} - 72 \, d^{6} g^{4} - {\left (2 \, e^{6} f^{4} - 12 \, d e^{5} f^{3} g + 42 \, d^{2} e^{4} f^{2} g^{2} + 128 \, d^{3} e^{3} f g^{3} + 117 \, d^{4} e^{2} g^{4}\right )} x^{2} + 3 \, {\left (2 \, d e^{5} f^{4} - 12 \, d^{2} e^{4} f^{3} g + 12 \, d^{3} e^{3} f^{2} g^{2} + 68 \, d^{4} e^{2} f g^{3} + 57 \, d^{5} e g^{4}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{3} e^{8} x^{3} - 3 \, d^{4} e^{7} x^{2} + 3 \, d^{5} e^{6} x - d^{6} e^{5}\right )}} \] Input:
integrate((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
Output:
-1/15*(7*d^3*e^4*f^4 - 12*d^4*e^3*f^3*g + 12*d^5*e^2*f^2*g^2 + 88*d^6*e*f* g^3 + 72*d^7*g^4 - (7*e^7*f^4 - 12*d*e^6*f^3*g + 12*d^2*e^5*f^2*g^2 + 88*d ^3*e^4*f*g^3 + 72*d^4*e^3*g^4)*x^3 + 3*(7*d*e^6*f^4 - 12*d^2*e^5*f^3*g + 1 2*d^3*e^4*f^2*g^2 + 88*d^4*e^3*f*g^3 + 72*d^5*e^2*g^4)*x^2 - 3*(7*d^2*e^5* f^4 - 12*d^3*e^4*f^3*g + 12*d^4*e^3*f^2*g^2 + 88*d^5*e^2*f*g^3 + 72*d^6*e* g^4)*x + 30*(4*d^6*e*f*g^3 + 3*d^7*g^4 - (4*d^3*e^4*f*g^3 + 3*d^4*e^3*g^4) *x^3 + 3*(4*d^4*e^3*f*g^3 + 3*d^5*e^2*g^4)*x^2 - 3*(4*d^5*e^2*f*g^3 + 3*d^ 6*e*g^4)*x)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*d^3*e^3*g^4*x^ 3 - 7*d^2*e^4*f^4 + 12*d^3*e^3*f^3*g - 12*d^4*e^2*f^2*g^2 - 88*d^5*e*f*g^3 - 72*d^6*g^4 - (2*e^6*f^4 - 12*d*e^5*f^3*g + 42*d^2*e^4*f^2*g^2 + 128*d^3 *e^3*f*g^3 + 117*d^4*e^2*g^4)*x^2 + 3*(2*d*e^5*f^4 - 12*d^2*e^4*f^3*g + 12 *d^3*e^3*f^2*g^2 + 68*d^4*e^2*f*g^3 + 57*d^5*e*g^4)*x)*sqrt(-e^2*x^2 + d^2 ))/(d^3*e^8*x^3 - 3*d^4*e^7*x^2 + 3*d^5*e^6*x - d^6*e^5)
\[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((e*x+d)**3*(g*x+f)**4/(-e**2*x**2+d**2)**(7/2),x)
Output:
Integral((d + e*x)**3*(f + g*x)**4/(-(-d + e*x)*(d + e*x))**(7/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (199) = 398\).
Time = 0.15 (sec) , antiderivative size = 1190, normalized size of antiderivative = 5.53 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
Output:
-e*g^4*x^6/(-e^2*x^2 + d^2)^(5/2) + 6*d^2*g^4*x^4/((-e^2*x^2 + d^2)^(5/2)* e) - 8*d^4*g^4*x^2/((-e^2*x^2 + d^2)^(5/2)*e^3) + 1/5*d*f^4*x/(-e^2*x^2 + d^2)^(5/2) + 1/15*(4*e^3*f*g^3 + 3*d*e^2*g^4)*x*(15*x^4/((-e^2*x^2 + d^2)^ (5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) + 3/5*d^2*f^4/((-e^2*x^2 + d^2)^(5/2)*e) + 4/5*d^3*f^3*g/ ((-e^2*x^2 + d^2)^(5/2)*e^2) + 16/5*d^6*g^4/((-e^2*x^2 + d^2)^(5/2)*e^5) + 4/15*f^4*x/((-e^2*x^2 + d^2)^(3/2)*d) + 8/15*f^4*x/(sqrt(-e^2*x^2 + d^2)* d^3) - 1/3*(4*e^3*f*g^3 + 3*d*e^2*g^4)*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^ 2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e^2 + 3*(2*e^3*f^2*g^2 + 4*d*e^2* f*g^3 + d^2*e*g^4)*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) + 1/2*(4*e^3*f^3*g + 1 8*d*e^2*f^2*g^2 + 12*d^2*e*f*g^3 + d^3*g^4)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^ 2) - 4*(2*e^3*f^2*g^2 + 4*d*e^2*f*g^3 + d^2*e*g^4)*d^2*x^2/((-e^2*x^2 + d^ 2)^(5/2)*e^4) + 1/3*(e^3*f^4 + 12*d*e^2*f^3*g + 18*d^2*e*f^2*g^2 + 4*d^3*f *g^3)*x^2/((-e^2*x^2 + d^2)^(5/2)*e^2) - 3/10*(4*e^3*f^3*g + 18*d*e^2*f^2* g^2 + 12*d^2*e*f*g^3 + d^3*g^4)*d^2*x/((-e^2*x^2 + d^2)^(5/2)*e^4) + 3/5*( d*e^2*f^4 + 4*d^2*e*f^3*g + 2*d^3*f^2*g^2)*x/((-e^2*x^2 + d^2)^(5/2)*e^2) + 8/5*(2*e^3*f^2*g^2 + 4*d*e^2*f*g^3 + d^2*e*g^4)*d^4/((-e^2*x^2 + d^2)^(5 /2)*e^6) - 2/15*(e^3*f^4 + 12*d*e^2*f^3*g + 18*d^2*e*f^2*g^2 + 4*d^3*f*g^3 )*d^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 4/15*(4*e^3*f*g^3 + 3*d*e^2*g^4)*d^2* x/((-e^2*x^2 + d^2)^(3/2)*e^6) + 1/10*(4*e^3*f^3*g + 18*d*e^2*f^2*g^2 +...
Leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (199) = 398\).
Time = 0.17 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.52 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Output:
sqrt(-e^2*x^2 + d^2)*g^4/e^5 - (4*e*f*g^3 + 3*d*g^4)*arcsin(e*x/d)*sgn(d)* sgn(e)/(e^4*abs(e)) + 2/15*(7*e^4*f^4 - 12*d*e^3*f^3*g + 12*d^2*e^2*f^2*g^ 2 + 88*d^3*e*f*g^3 + 57*d^4*g^4 - 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e ^2*f^4/x + 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e*f^3*g/x - 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*f^2*g^2/x - 380*(d*e + sqrt(-e^2*x^2 + d^ 2)*abs(e))*d^3*f*g^3/(e*x) - 240*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^4*g ^4/(e^2*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*f^4/x^2 - 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f^3*g/(e*x^2) + 120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2*f^2*g^2/(e^2*x^2) + 580*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))^2*d^3*f*g^3/(e^3*x^2) + 360*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d ^4*g^4/(e^4*x^2) - 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^4/(e^2*x^3) + 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*f^3*g/(e^3*x^3) - 300*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3*f*g^3/(e^5*x^3) - 210*(d*e + sqrt(-e^2* x^2 + d^2)*abs(e))^3*d^4*g^4/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*ab s(e))^4*f^4/(e^4*x^4) + 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*f*g^3 /(e^7*x^4) + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^4*g^4/(e^8*x^4))/( d^3*e^4*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^4\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \] Input:
int(((f + g*x)^4*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)
Output:
int(((f + g*x)^4*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)
Time = 0.25 (sec) , antiderivative size = 1019, normalized size of antiderivative = 4.74 \[ \int \frac {(d+e x)^3 (f+g x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3*(g*x+f)^4/(-e^2*x^2+d^2)^(7/2),x)
Output:
( - 45*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**6*g**4 - 60*sqrt(d**2 - e** 2*x**2)*asin((e*x)/d)*d**5*e*f*g**3 + 90*sqrt(d**2 - e**2*x**2)*asin((e*x) /d)*d**5*e*g**4*x + 120*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**4*e**2*f*g **3*x - 45*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**4*e**2*g**4*x**2 - 60*s qrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**3*e**3*f*g**3*x**2 + 45*asin((e*x)/ d)*d**7*g**4 + 60*asin((e*x)/d)*d**6*e*f*g**3 - 135*asin((e*x)/d)*d**6*e*g **4*x - 180*asin((e*x)/d)*d**5*e**2*f*g**3*x + 135*asin((e*x)/d)*d**5*e**2 *g**4*x**2 + 180*asin((e*x)/d)*d**4*e**3*f*g**3*x**2 - 45*asin((e*x)/d)*d* *4*e**3*g**4*x**3 - 60*asin((e*x)/d)*d**3*e**4*f*g**3*x**3 - 18*sqrt(d**2 - e**2*x**2)*d**6*g**4 - 24*sqrt(d**2 - e**2*x**2)*d**5*e*f*g**3 + 63*sqrt (d**2 - e**2*x**2)*d**5*e*g**4*x + 76*sqrt(d**2 - e**2*x**2)*d**4*e**2*f*g **3*x - 63*sqrt(d**2 - e**2*x**2)*d**4*e**2*g**4*x**2 + 12*sqrt(d**2 - e** 2*x**2)*d**3*e**3*f**2*g**2*x - 64*sqrt(d**2 - e**2*x**2)*d**3*e**3*f*g**3 *x**2 + 15*sqrt(d**2 - e**2*x**2)*d**3*e**3*g**4*x**3 - 6*sqrt(d**2 - e**2 *x**2)*d**2*e**4*f**4 - 12*sqrt(d**2 - e**2*x**2)*d**2*e**4*f**3*g*x - 30* sqrt(d**2 - e**2*x**2)*d**2*e**4*f**2*g**2*x**2 + 4*sqrt(d**2 - e**2*x**2) *d*e**5*f**4*x - sqrt(d**2 - e**2*x**2)*e**6*f**4*x**2 + 18*d**7*g**4 + 24 *d**6*e*f*g**3 + 63*d**6*e*g**4*x + 76*d**5*e**2*f*g**3*x - 216*d**5*e**2* g**4*x**2 + 12*d**4*e**3*f**2*g**2*x - 268*d**4*e**3*f*g**3*x**2 + 156*d** 4*e**3*g**4*x**3 + 6*d**3*e**4*f**4 - 12*d**3*e**4*f**3*g*x - 30*d**3*e...