\(\int \frac {(c^2-d^2 x^2)^{2/3}}{(c+d x)^{7/3}} \, dx\) [283]

Optimal result

Integrand size = 26, antiderivative size = 224 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{2 d (c+d x)^{4/3}}-\frac {\sqrt {3} \left (c^2-d^2 x^2\right )^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c-d x}}{\sqrt {3} \sqrt [3]{c+d x}}\right )}{d (c-d x)^{2/3} (c+d x)^{2/3}}-\frac {\left (c^2-d^2 x^2\right )^{2/3} \log (c+d x)}{2 d (c-d x)^{2/3} (c+d x)^{2/3}}-\frac {3 \left (c^2-d^2 x^2\right )^{2/3} \log \left (1+\frac {\sqrt [3]{c-d x}}{\sqrt [3]{c+d x}}\right )}{2 d (c-d x)^{2/3} (c+d x)^{2/3}} \] Output:

-3/2*(-d^2*x^2+c^2)^(2/3)/d/(d*x+c)^(4/3)+3^(1/2)*(-d^2*x^2+c^2)^(2/3)*arc 
tan(-1/3*3^(1/2)+2/3*(-d*x+c)^(1/3)*3^(1/2)/(d*x+c)^(1/3))/d/(-d*x+c)^(2/3 
)/(d*x+c)^(2/3)-1/2*(-d^2*x^2+c^2)^(2/3)*ln(d*x+c)/d/(-d*x+c)^(2/3)/(d*x+c 
)^(2/3)-3/2*(-d^2*x^2+c^2)^(2/3)*ln(1+(-d*x+c)^(1/3)/(d*x+c)^(1/3))/d/(-d* 
x+c)^(2/3)/(d*x+c)^(2/3)
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.77 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\frac {-\frac {3 \left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{4/3}}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (c+d x)^{2/3}}{(c+d x)^{2/3}-2 \sqrt [3]{c^2-d^2 x^2}}\right )-2 \log \left (d \left ((c+d x)^{2/3}+\sqrt [3]{c^2-d^2 x^2}\right )\right )+\log \left ((c+d x)^{4/3}-(c+d x)^{2/3} \sqrt [3]{c^2-d^2 x^2}+\left (c^2-d^2 x^2\right )^{2/3}\right )}{2 d} \] Input:

Integrate[(c^2 - d^2*x^2)^(2/3)/(c + d*x)^(7/3),x]
 

Output:

((-3*(c^2 - d^2*x^2)^(2/3))/(c + d*x)^(4/3) + 2*Sqrt[3]*ArcTan[(Sqrt[3]*(c 
 + d*x)^(2/3))/((c + d*x)^(2/3) - 2*(c^2 - d^2*x^2)^(1/3))] - 2*Log[d*((c 
+ d*x)^(2/3) + (c^2 - d^2*x^2)^(1/3))] + Log[(c + d*x)^(4/3) - (c + d*x)^( 
2/3)*(c^2 - d^2*x^2)^(1/3) + (c^2 - d^2*x^2)^(2/3)])/(2*d)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {474, 473, 57, 72}

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.

\(\displaystyle \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {\sqrt [3]{\frac {d x}{c}+1} \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{\left (\frac {d x}{c}+1\right )^{7/3}}dx}{c^2 \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 57

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

\(\Big \downarrow \) 72

\(\displaystyle \frac {\left (c^2-d^2 x^2\right )^{5/3} \left (-\left (c^2 \left (\frac {\sqrt {3} \sqrt [3]{c} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c^2-c d x}}{\sqrt {3} c^{2/3} \sqrt [3]{\frac {d x}{c}+1}}\right )}{d}+\frac {3 \sqrt [3]{c} \log \left (\frac {\sqrt [3]{c^2-c d x}}{c^{2/3} \sqrt [3]{\frac {d x}{c}+1}}+1\right )}{2 d}+\frac {\sqrt [3]{c} \log \left (\frac {d x}{c}+1\right )}{2 d}\right )\right )-\frac {3 c \left (c^2-c d x\right )^{2/3}}{2 d \left (\frac {d x}{c}+1\right )^{2/3}}\right )}{c^2 \sqrt [3]{c+d x} \left (\frac {d x}{c}+1\right )^{4/3} \left (c^2-c d x\right )^{5/3}}\)

Input:

Int[(c^2 - d^2*x^2)^(2/3)/(c + d*x)^(7/3),x]
 

Output:

((c^2 - d^2*x^2)^(5/3)*((-3*c*(c^2 - c*d*x)^(2/3))/(2*d*(1 + (d*x)/c)^(2/3 
)) - c^2*((Sqrt[3]*c^(1/3)*ArcTan[1/Sqrt[3] - (2*(c^2 - c*d*x)^(1/3))/(Sqr 
t[3]*c^(2/3)*(1 + (d*x)/c)^(1/3))])/d + (c^(1/3)*Log[1 + (d*x)/c])/(2*d) + 
 (3*c^(1/3)*Log[1 + (c^2 - c*d*x)^(1/3)/(c^(2/3)*(1 + (d*x)/c)^(1/3))])/(2 
*d))))/(c^2*(c + d*x)^(1/3)*(1 + (d*x)/c)^(4/3)*(c^2 - c*d*x)^(5/3))
 

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 72

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> 
 With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* 
x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* 
x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F 
reeQ[{a, b, c, d}, x] && NegQ[d/b]
 

rule 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 474

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(1 + d 
*(x/c))^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + 
 a*d^2, 0] &&  !(IntegerQ[n] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}{\left (d x +c \right )^{\frac {7}{3}}}d x\]

Input:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x)
 

Output:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.32 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {3} {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \sqrt {3} {\left (d^{2} x^{2} - c^{2}\right )}}{3 \, {\left (d^{2} x^{2} - c^{2}\right )}}\right ) + {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (\frac {d^{2} x^{2} - c^{2} - {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {4}{3}} + {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d^{2} x^{2} - c^{2}}\right ) - 2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (-\frac {d^{2} x^{2} - c^{2} - {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d^{2} x^{2} - c^{2}}\right ) - 3 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x, algorithm="fricas")
 

Output:

1/2*(2*sqrt(3)*(d^2*x^2 + 2*c*d*x + c^2)*arctan(1/3*(2*sqrt(3)*(-d^2*x^2 + 
 c^2)^(2/3)*(d*x + c)^(2/3) + sqrt(3)*(d^2*x^2 - c^2))/(d^2*x^2 - c^2)) + 
(d^2*x^2 + 2*c*d*x + c^2)*log((d^2*x^2 - c^2 - (-d^2*x^2 + c^2)^(1/3)*(d*x 
 + c)^(4/3) + (-d^2*x^2 + c^2)^(2/3)*(d*x + c)^(2/3))/(d^2*x^2 - c^2)) - 2 
*(d^2*x^2 + 2*c*d*x + c^2)*log(-(d^2*x^2 - c^2 - (-d^2*x^2 + c^2)^(2/3)*(d 
*x + c)^(2/3))/(d^2*x^2 - c^2)) - 3*(-d^2*x^2 + c^2)^(2/3)*(d*x + c)^(2/3) 
)/(d^3*x^2 + 2*c*d^2*x + c^2*d)
 

Sympy [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\int \frac {\left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {2}{3}}}{\left (c + d x\right )^{\frac {7}{3}}}\, dx \] Input:

integrate((-d**2*x**2+c**2)**(2/3)/(d*x+c)**(7/3),x)
 

Output:

Integral((-(-c + d*x)*(c + d*x))**(2/3)/(c + d*x)**(7/3), x)
 

Maxima [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\int { \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2 + c^2)^(2/3)/(d*x + c)^(7/3), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.46 \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - 3 \, {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {2}{3}} + \log \left ({\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {2}{3}} - {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, \log \left ({\left | {\left (\frac {2 \, c}{d x + c} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right )}{2 \, d} \] Input:

integrate((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x, algorithm="giac")
 

Output:

1/2*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(2*c/(d*x + c) - 1)^(1/3) - 1)) - 3*( 
2*c/(d*x + c) - 1)^(2/3) + log((2*c/(d*x + c) - 1)^(2/3) - (2*c/(d*x + c) 
- 1)^(1/3) + 1) - 2*log(abs((2*c/(d*x + c) - 1)^(1/3) + 1)))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\int \frac {{\left (c^2-d^2\,x^2\right )}^{2/3}}{{\left (c+d\,x\right )}^{7/3}} \,d x \] Input:

int((c^2 - d^2*x^2)^(2/3)/(c + d*x)^(7/3),x)
 

Output:

int((c^2 - d^2*x^2)^(2/3)/(c + d*x)^(7/3), x)
 

Reduce [F]

\[ \int \frac {\left (c^2-d^2 x^2\right )^{2/3}}{(c+d x)^{7/3}} \, dx=\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {2}{3}}}{\left (d x +c \right )^{\frac {1}{3}} c^{2}+2 \left (d x +c \right )^{\frac {1}{3}} c d x +\left (d x +c \right )^{\frac {1}{3}} d^{2} x^{2}}d x \] Input:

int((-d^2*x^2+c^2)^(2/3)/(d*x+c)^(7/3),x)
 

Output:

int((c**2 - d**2*x**2)**(2/3)/((c + d*x)**(1/3)*c**2 + 2*(c + d*x)**(1/3)* 
c*d*x + (c + d*x)**(1/3)*d**2*x**2),x)