3.611 \(\int \frac {(a+c x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=196 \[ \frac {2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac {8 c^2 d \sqrt {d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac {6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt {d+e x}}+\frac {4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac {12 c^3 d (d+e x)^{5/2}}{5 e^7} \]

[Out]

-2/5*(a*e^2+c*d^2)^3/e^7/(e*x+d)^(5/2)+4*c*d*(a*e^2+c*d^2)^2/e^7/(e*x+d)^(3/2)+2*c^2*(a*e^2+5*c*d^2)*(e*x+d)^(
3/2)/e^7-12/5*c^3*d*(e*x+d)^(5/2)/e^7+2/7*c^3*(e*x+d)^(7/2)/e^7-6*c*(a*e^2+c*d^2)*(a*e^2+5*c*d^2)/e^7/(e*x+d)^
(1/2)-8*c^2*d*(3*a*e^2+5*c*d^2)*(e*x+d)^(1/2)/e^7

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Rubi [A]  time = 0.08, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {697} \[ \frac {2 c^2 (d+e x)^{3/2} \left (a e^2+5 c d^2\right )}{e^7}-\frac {8 c^2 d \sqrt {d+e x} \left (3 a e^2+5 c d^2\right )}{e^7}-\frac {6 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 \sqrt {d+e x}}+\frac {4 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {2 \left (a e^2+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}-\frac {12 c^3 d (d+e x)^{5/2}}{5 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^3/(d + e*x)^(7/2),x]

[Out]

(-2*(c*d^2 + a*e^2)^3)/(5*e^7*(d + e*x)^(5/2)) + (4*c*d*(c*d^2 + a*e^2)^2)/(e^7*(d + e*x)^(3/2)) - (6*c*(c*d^2
+ a*e^2)*(5*c*d^2 + a*e^2))/(e^7*Sqrt[d + e*x]) - (8*c^2*d*(5*c*d^2 + 3*a*e^2)*Sqrt[d + e*x])/e^7 + (2*c^2*(5
*c*d^2 + a*e^2)*(d + e*x)^(3/2))/e^7 - (12*c^3*d*(d + e*x)^(5/2))/(5*e^7) + (2*c^3*(d + e*x)^(7/2))/(7*e^7)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{7/2}}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^{5/2}}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^{3/2}}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 \sqrt {d+e x}}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) \sqrt {d+e x}}{e^6}-\frac {6 c^3 d (d+e x)^{3/2}}{e^6}+\frac {c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac {2 \left (c d^2+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac {4 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 \sqrt {d+e x}}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) \sqrt {d+e x}}{e^7}+\frac {2 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^7}-\frac {12 c^3 d (d+e x)^{5/2}}{5 e^7}+\frac {2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 170, normalized size = 0.87 \[ -\frac {2 \left (7 a^3 e^6+7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a c^2 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a*c^2*e^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^
2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^4) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 40*d^2
*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)))/(35*e^7*(d + e*x)^(5/2))

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fricas [A]  time = 0.95, size = 234, normalized size = 1.19 \[ \frac {2 \, {\left (5 \, c^{3} e^{6} x^{6} - 12 \, c^{3} d e^{5} x^{5} - 1024 \, c^{3} d^{6} - 896 \, a c^{2} d^{4} e^{2} - 56 \, a^{2} c d^{2} e^{4} - 7 \, a^{3} e^{6} + 5 \, {\left (8 \, c^{3} d^{2} e^{4} + 7 \, a c^{2} e^{6}\right )} x^{4} - 40 \, {\left (8 \, c^{3} d^{3} e^{3} + 7 \, a c^{2} d e^{5}\right )} x^{3} - 15 \, {\left (128 \, c^{3} d^{4} e^{2} + 112 \, a c^{2} d^{2} e^{4} + 7 \, a^{2} c e^{6}\right )} x^{2} - 20 \, {\left (128 \, c^{3} d^{5} e + 112 \, a c^{2} d^{3} e^{3} + 7 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*c^3*e^6*x^6 - 12*c^3*d*e^5*x^5 - 1024*c^3*d^6 - 896*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 7*a^3*e^6 + 5*(
8*c^3*d^2*e^4 + 7*a*c^2*e^6)*x^4 - 40*(8*c^3*d^3*e^3 + 7*a*c^2*d*e^5)*x^3 - 15*(128*c^3*d^4*e^2 + 112*a*c^2*d^
2*e^4 + 7*a^2*c*e^6)*x^2 - 20*(128*c^3*d^5*e + 112*a*c^2*d^3*e^3 + 7*a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^10*x^3 +
3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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giac [A]  time = 0.28, size = 251, normalized size = 1.28 \[ \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} e^{42} - 42 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d e^{42} + 175 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt {x e + d} c^{3} d^{3} e^{42} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} e^{44} - 420 \, \sqrt {x e + d} a c^{2} d e^{44}\right )} e^{\left (-49\right )} - \frac {2 \, {\left (75 \, {\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \, {\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} + 90 \, {\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \, {\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 15 \, {\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \, {\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \, {\left (x e + d\right )}^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*e^42 - 42*(x*e + d)^(5/2)*c^3*d*e^42 + 175*(x*e + d)^(3/2)*c^3*d^2*e^42 - 700*sqrt
(x*e + d)*c^3*d^3*e^42 + 35*(x*e + d)^(3/2)*a*c^2*e^44 - 420*sqrt(x*e + d)*a*c^2*d*e^44)*e^(-49) - 2/5*(75*(x*
e + d)^2*c^3*d^4 - 10*(x*e + d)*c^3*d^5 + c^3*d^6 + 90*(x*e + d)^2*a*c^2*d^2*e^2 - 20*(x*e + d)*a*c^2*d^3*e^2
+ 3*a*c^2*d^4*e^2 + 15*(x*e + d)^2*a^2*c*e^4 - 10*(x*e + d)*a^2*c*d*e^4 + 3*a^2*c*d^2*e^4 + a^3*e^6)*e^(-7)/(x
*e + d)^(5/2)

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maple [A]  time = 0.06, size = 205, normalized size = 1.05 \[ -\frac {2 \left (-5 c^{3} x^{6} e^{6}+12 c^{3} d \,e^{5} x^{5}-35 a \,c^{2} e^{6} x^{4}-40 c^{3} d^{2} e^{4} x^{4}+280 a \,c^{2} d \,e^{5} x^{3}+320 c^{3} d^{3} e^{3} x^{3}+105 a^{2} c \,e^{6} x^{2}+1680 a \,c^{2} d^{2} e^{4} x^{2}+1920 c^{3} d^{4} e^{2} x^{2}+140 a^{2} c d \,e^{5} x +2240 a \,c^{2} d^{3} e^{3} x +2560 c^{3} d^{5} e x +7 e^{6} a^{3}+56 a^{2} c \,d^{2} e^{4}+896 a \,c^{2} d^{4} e^{2}+1024 c^{3} d^{6}\right )}{35 \left (e x +d \right )^{\frac {5}{2}} e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35/(e*x+d)^(5/2)*(-5*c^3*e^6*x^6+12*c^3*d*e^5*x^5-35*a*c^2*e^6*x^4-40*c^3*d^2*e^4*x^4+280*a*c^2*d*e^5*x^3+3
20*c^3*d^3*e^3*x^3+105*a^2*c*e^6*x^2+1680*a*c^2*d^2*e^4*x^2+1920*c^3*d^4*e^2*x^2+140*a^2*c*d*e^5*x+2240*a*c^2*
d^3*e^3*x+2560*c^3*d^5*e*x+7*a^3*e^6+56*a^2*c*d^2*e^4+896*a*c^2*d^4*e^2+1024*c^3*d^6)/e^7

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maxima [A]  time = 1.41, size = 215, normalized size = 1.10 \[ \frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d + 35 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 140 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \sqrt {e x + d}}{e^{6}} - \frac {7 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 15 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} {\left (e x + d\right )}^{2} - 10 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{6}}\right )}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*((5*(e*x + d)^(7/2)*c^3 - 42*(e*x + d)^(5/2)*c^3*d + 35*(5*c^3*d^2 + a*c^2*e^2)*(e*x + d)^(3/2) - 140*(5*
c^3*d^3 + 3*a*c^2*d*e^2)*sqrt(e*x + d))/e^6 - 7*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6 + 15*(5
*c^3*d^4 + 6*a*c^2*d^2*e^2 + a^2*c*e^4)*(e*x + d)^2 - 10*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*(e*x + d))/
((e*x + d)^(5/2)*e^6))/e

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mupad [B]  time = 0.06, size = 222, normalized size = 1.13 \[ \frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7}-\frac {{\left (d+e\,x\right )}^2\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )-\left (d+e\,x\right )\,\left (4\,a^2\,c\,d\,e^4+8\,a\,c^2\,d^3\,e^2+4\,c^3\,d^5\right )+\frac {2\,a^3\,e^6}{5}+\frac {2\,c^3\,d^6}{5}+\frac {6\,a\,c^2\,d^4\,e^2}{5}+\frac {6\,a^2\,c\,d^2\,e^4}{5}}{e^7\,{\left (d+e\,x\right )}^{5/2}}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,\sqrt {d+e\,x}}{e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3/(d + e*x)^(7/2),x)

[Out]

((30*c^3*d^2 + 6*a*c^2*e^2)*(d + e*x)^(3/2))/(3*e^7) + (2*c^3*(d + e*x)^(7/2))/(7*e^7) - ((d + e*x)^2*(30*c^3*
d^4 + 6*a^2*c*e^4 + 36*a*c^2*d^2*e^2) - (d + e*x)*(4*c^3*d^5 + 8*a*c^2*d^3*e^2 + 4*a^2*c*d*e^4) + (2*a^3*e^6)/
5 + (2*c^3*d^6)/5 + (6*a*c^2*d^4*e^2)/5 + (6*a^2*c*d^2*e^4)/5)/(e^7*(d + e*x)^(5/2)) - ((40*c^3*d^3 + 24*a*c^2
*d*e^2)*(d + e*x)^(1/2))/e^7 - (12*c^3*d*(d + e*x)^(5/2))/(5*e^7)

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sympy [A]  time = 78.23, size = 197, normalized size = 1.01 \[ - \frac {12 c^{3} d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{7}} + \frac {2 c^{3} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{7}} + \frac {4 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac {3}{2}}} - \frac {6 c \left (a e^{2} + c d^{2}\right ) \left (a e^{2} + 5 c d^{2}\right )}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (6 a c^{2} e^{2} + 30 c^{3} d^{2}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (- 24 a c^{2} d e^{2} - 40 c^{3} d^{3}\right )}{e^{7}} - \frac {2 \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*c**3*d*(d + e*x)**(5/2)/(5*e**7) + 2*c**3*(d + e*x)**(7/2)/(7*e**7) + 4*c*d*(a*e**2 + c*d**2)**2/(e**7*(d
+ e*x)**(3/2)) - 6*c*(a*e**2 + c*d**2)*(a*e**2 + 5*c*d**2)/(e**7*sqrt(d + e*x)) + (d + e*x)**(3/2)*(6*a*c**2*e
**2 + 30*c**3*d**2)/(3*e**7) + sqrt(d + e*x)*(-24*a*c**2*d*e**2 - 40*c**3*d**3)/e**7 - 2*(a*e**2 + c*d**2)**3/
(5*e**7*(d + e*x)**(5/2))

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