Integrand size = 21, antiderivative size = 426 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.32 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {753, 847, 858, 733, 435, 430} \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {e \sqrt {a+c x^2} \sqrt {d+e x} \left (3 c d^2-5 a e^2\right )}{3 a c^2}-\frac {(d+e x)^{5/2} (a e-c d x)}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {a+c x^2} (d+e x)^{3/2}}{a c} \]
[In]
[Out]
Rule 430
Rule 435
Rule 733
Rule 753
Rule 847
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {5 a e^2}{2}-\frac {5}{2} c d e x\right )}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}+\frac {2 \int \frac {\sqrt {d+e x} \left (10 a c d e^2-\frac {5}{4} c e \left (3 c d^2-5 a e^2\right ) x\right )}{\sqrt {a+c x^2}} \, dx}{5 a c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}+\frac {4 \int \frac {\frac {5}{8} a c e^2 \left (27 c d^2-5 a e^2\right )-\frac {5}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 a c^3} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {\left (d \left (3 c d^2-29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{6 a c}+\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{6 a c^2} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {\left (d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = -\frac {(a e-c d x) (d+e x)^{5/2}}{a c \sqrt {a+c x^2}}-\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}{3 a c^2}-\frac {d e (d+e x)^{3/2} \sqrt {a+c x^2}}{a c}-\frac {d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} c^{5/2} \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 24.70 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {10 a e^3}{c^2}+\frac {6 d^3 x}{a}+\frac {2 e \left (-9 d^2-9 d e x+2 e^2 x^2\right )}{c}+\frac {2 \left (-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (3 c d^2-29 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} d \left (3 i c^{3/2} d^3-3 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+29 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (3 c^{3/2} d^3+27 i \sqrt {a} c d^2 e-29 a \sqrt {c} d e^2-5 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{a c^2 e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{6 \sqrt {a+c x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(793\) vs. \(2(352)=704\).
Time = 5.94 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.86
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) d x}{2 c^{2} a}-\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 e^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 c^{2}}+\frac {2 \left (-\frac {e^{2} \left (e^{2} a -6 c \,d^{2}\right )}{c^{2}}+\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}}{c^{2} a}-\frac {e^{2} \left (e^{2} a -3 c \,d^{2}\right )}{2 c^{2}}+\frac {d^{2} \left (3 e^{2} a -c \,d^{2}\right )}{c a}-\frac {a \,e^{4}}{3 c^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {10 d \,e^{3}}{3 c}+\frac {\left (3 e^{2} a -c \,d^{2}\right ) d e}{2 a c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(794\) |
default | \(\text {Expression too large to display}\) | \(1362\) |
risch | \(\text {Expression too large to display}\) | \(1487\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, a c^{2} d^{4} + 52 \, a^{2} c d^{2} e^{2} - 15 \, a^{3} e^{4} + {\left (3 \, c^{3} d^{4} + 52 \, a c^{2} d^{2} e^{2} - 15 \, a^{2} c e^{4}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (3 \, a c^{2} d^{3} e - 29 \, a^{2} c d e^{3} + {\left (3 \, c^{3} d^{3} e - 29 \, a c^{2} d e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (2 \, a c^{2} e^{4} x^{2} - 9 \, a c^{2} d^{2} e^{2} + 5 \, a^{2} c e^{4} + 3 \, {\left (c^{3} d^{3} e - 3 \, a c^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{9 \, {\left (a c^{4} e x^{2} + a^{2} c^{3} e\right )}} \]
[In]
[Out]
\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
[In]
[Out]