3.6.98 \(\int (d+e x)^{5/2} (a+c x^2)^2 \, dx\) [598]

Optimal. Leaf size=127 \[ \frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

[Out]

2/7*(a*e^2+c*d^2)^2*(e*x+d)^(7/2)/e^5-8/9*c*d*(a*e^2+c*d^2)*(e*x+d)^(9/2)/e^5+4/11*c*(a*e^2+3*c*d^2)*(e*x+d)^(
11/2)/e^5-8/13*c^2*d*(e*x+d)^(13/2)/e^5+2/15*c^2*(e*x+d)^(15/2)/e^5

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Rubi [A]
time = 0.04, antiderivative size = 127, 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 = {711} \begin {gather*} \frac {4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac {8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2)^2,x]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5
)

Rule 711

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 (d+e x)^{5/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^4}-\frac {4 c^2 d (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac {2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac {8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 96, normalized size = 0.76 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (6435 a^2 e^4+130 a c e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(a + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(6435*a^2*e^4 + 130*a*c*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + c^2*(128*d^4 - 448*d^3*e*x +
1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5)

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Maple [A]
time = 0.42, size = 108, normalized size = 0.85

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} e^{4} x^{4}-1848 c^{2} d \,x^{3} e^{3}+8190 a c \,e^{4} x^{2}+1008 d^{2} e^{2} x^{2} c^{2}-3640 a c d \,e^{3} x -448 c^{2} d^{3} e x +6435 a^{2} e^{4}+1040 a c \,d^{2} e^{2}+128 c^{2} d^{4}\right )}{45045 e^{5}}\)	\(106\)
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(108\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 d^{2} c^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {8 \left (e^{2} a +c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(108\)
trager	\(\frac {2 \left (3003 c^{2} e^{7} x^{7}+7161 c^{2} d \,e^{6} x^{6}+8190 e^{7} a c \,x^{5}+4473 e^{5} d^{2} c^{2} x^{5}+20930 a c d \,e^{6} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 e^{7} a^{2} x^{3}+14690 e^{5} d^{2} a c \,x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+390 a c \,d^{3} e^{4} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x -520 a c \,d^{4} e^{3} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}+1040 a c \,d^{5} e^{2}+128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(225\)
risch	\(\frac {2 \left (3003 c^{2} e^{7} x^{7}+7161 c^{2} d \,e^{6} x^{6}+8190 e^{7} a c \,x^{5}+4473 e^{5} d^{2} c^{2} x^{5}+20930 a c d \,e^{6} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 e^{7} a^{2} x^{3}+14690 e^{5} d^{2} a c \,x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+390 a c \,d^{3} e^{4} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x -520 a c \,d^{4} e^{3} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}+1040 a c \,d^{5} e^{2}+128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/15*c^2*(e*x+d)^(15/2)-4/13*c^2*d*(e*x+d)^(13/2)+1/11*(2*(a*e^2+c*d^2)*c+4*d^2*c^2)*(e*x+d)^(11/2)-4/9
*(a*e^2+c*d^2)*c*d*(e*x+d)^(9/2)+1/7*(a*e^2+c*d^2)^2*(e*x+d)^(7/2))

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Maxima [A]
time = 0.30, size = 113, normalized size = 0.89 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, {\left (x e + d\right )}^{\frac {15}{2}} c^{2} - 13860 \, {\left (x e + d\right )}^{\frac {13}{2}} c^{2} d + 8190 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} {\left (x e + d\right )}^{\frac {11}{2}} - 20020 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} {\left (x e + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {7}{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+a)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(x*e + d)^(15/2)*c^2 - 13860*(x*e + d)^(13/2)*c^2*d + 8190*(3*c^2*d^2 + a*c*e^2)*(x*e + d)^(11/2
) - 20020*(c^2*d^3 + a*c*d*e^2)*(x*e + d)^(9/2) + 6435*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(x*e + d)^(7/2))*e^
(-5)

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Fricas [A]
time = 3.34, size = 206, normalized size = 1.62 \begin {gather*} -\frac {2}{45045} \, {\left (64 \, c^{2} d^{6} x e - 128 \, c^{2} d^{7} - 39 \, {\left (77 \, c^{2} x^{7} + 210 \, a c x^{5} + 165 \, a^{2} x^{3}\right )} e^{7} - {\left (7161 \, c^{2} d x^{6} + 20930 \, a c d x^{4} + 19305 \, a^{2} d x^{2}\right )} e^{6} - {\left (4473 \, c^{2} d^{2} x^{5} + 14690 \, a c d^{2} x^{3} + 19305 \, a^{2} d^{2} x\right )} e^{5} - 5 \, {\left (7 \, c^{2} d^{3} x^{4} + 78 \, a c d^{3} x^{2} + 1287 \, a^{2} d^{3}\right )} e^{4} + 40 \, {\left (c^{2} d^{4} x^{3} + 13 \, a c d^{4} x\right )} e^{3} - 16 \, {\left (3 \, c^{2} d^{5} x^{2} + 65 \, a c d^{5}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-2/45045*(64*c^2*d^6*x*e - 128*c^2*d^7 - 39*(77*c^2*x^7 + 210*a*c*x^5 + 165*a^2*x^3)*e^7 - (7161*c^2*d*x^6 + 2
0930*a*c*d*x^4 + 19305*a^2*d*x^2)*e^6 - (4473*c^2*d^2*x^5 + 14690*a*c*d^2*x^3 + 19305*a^2*d^2*x)*e^5 - 5*(7*c^
2*d^3*x^4 + 78*a*c*d^3*x^2 + 1287*a^2*d^3)*e^4 + 40*(c^2*d^4*x^3 + 13*a*c*d^4*x)*e^3 - 16*(3*c^2*d^5*x^2 + 65*
a*c*d^5)*e^2)*sqrt(x*e + d)*e^(-5)

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Sympy [A]
time = 9.94, size = 566, normalized size = 4.46 \begin {gather*} a^{2} d^{2} \left (\begin {cases} \sqrt {d} x & \text {for}\: e = 0 \\\frac {2 \left (d + e x\right )^{\frac {3}{2}}}{3 e} & \text {otherwise} \end {cases}\right ) + \frac {4 a^{2} d \left (- \frac {d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {2 a^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e} + \frac {4 a c d^{2} \left (\frac {d^{2} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {2 d \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{3}} + \frac {8 a c d \left (- \frac {d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {3 d^{2} \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {3 d \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}} + \frac {4 a c \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{3}} + \frac {2 c^{2} d^{2} \left (\frac {d^{4} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {4 d^{3} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {6 d^{2} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {4 d \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {\left (d + e x\right )^{\frac {11}{2}}}{11}\right )}{e^{5}} + \frac {4 c^{2} d \left (- \frac {d^{5} \left (d + e x\right )^{\frac {3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac {5}{2}} - \frac {10 d^{3} \left (d + e x\right )^{\frac {7}{2}}}{7} + \frac {10 d^{2} \left (d + e x\right )^{\frac {9}{2}}}{9} - \frac {5 d \left (d + e x\right )^{\frac {11}{2}}}{11} + \frac {\left (d + e x\right )^{\frac {13}{2}}}{13}\right )}{e^{5}} + \frac {2 c^{2} \left (\frac {d^{6} \left (d + e x\right )^{\frac {3}{2}}}{3} - \frac {6 d^{5} \left (d + e x\right )^{\frac {5}{2}}}{5} + \frac {15 d^{4} \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {20 d^{3} \left (d + e x\right )^{\frac {9}{2}}}{9} + \frac {15 d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11} - \frac {6 d \left (d + e x\right )^{\frac {13}{2}}}{13} + \frac {\left (d + e x\right )^{\frac {15}{2}}}{15}\right )}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(c*x**2+a)**2,x)

[Out]

a**2*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**2*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
4*a*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 8*a*c*d*(-d**3*(d +
e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 4*a*c*(d**4*(d
+ e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)*
*(11/2)/11)/e**3 + 2*c**2*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 4*c**2*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)*
*(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2
)/13)/e**5 + 2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**
3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (104) = 208\).
time = 3.86, size = 749, normalized size = 5.90 \begin {gather*} \frac {2}{45045} \, {\left (6006 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c d^{3} e^{\left (-2\right )} + 143 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{2} d^{3} e^{\left (-4\right )} + 7722 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c d^{2} e^{\left (-2\right )} + 195 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{2} d^{2} e^{\left (-4\right )} + 45045 \, \sqrt {x e + d} a^{2} d^{3} + 45045 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d^{2} + 858 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c d e^{\left (-2\right )} + 45 \, {\left (231 \, {\left (x e + d\right )}^{\frac {13}{2}} - 1638 \, {\left (x e + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {x e + d} d^{6}\right )} c^{2} d e^{\left (-4\right )} + 9009 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} d + 130 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} a c e^{\left (-2\right )} + 7 \, {\left (429 \, {\left (x e + d\right )}^{\frac {15}{2}} - 3465 \, {\left (x e + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (x e + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (x e + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {x e + d} d^{7}\right )} c^{2} e^{\left (-4\right )} + 1287 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(c*x^2+a)^2,x, algorithm="giac")

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*d^3*e^(-2) + 143*(35*(x*e
+ d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4
)*c^2*d^3*e^(-4) + 7722*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*
d^3)*a*c*d^2*e^(-2) + 195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e +
d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c^2*d^2*e^(-4) + 45045*sqrt(x*e + d)*a^2*d^3
+ 45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*d^2 + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378
*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*c*d*e^(-2) + 45*(231*(x*e + d)^(13/2
) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 -
6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*c^2*d*e^(-4) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2
)*d + 15*sqrt(x*e + d)*d^2)*a^2*d + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2
- 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*c*e^(-2) + 7*(429*(x*e + d)^
(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7
/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^2*e^(-4) + 1287*(5
*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2)*e^(-1)

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Mupad [B]
time = 0.35, size = 114, normalized size = 0.90 \begin {gather*} \frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}-\frac {\left (8\,c^2\,d^3+8\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {\left (12\,c^2\,d^2+4\,a\,c\,e^2\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {8\,c^2\,d\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2*(d + e*x)^(5/2),x)

[Out]

(2*c^2*(d + e*x)^(15/2))/(15*e^5) - ((8*c^2*d^3 + 8*a*c*d*e^2)*(d + e*x)^(9/2))/(9*e^5) + (2*(a*e^2 + c*d^2)^2
*(d + e*x)^(7/2))/(7*e^5) + ((12*c^2*d^2 + 4*a*c*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(d + e*x)^(13/2))/
(13*e^5)

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